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On the Optimization of the Methods for Solving Boundary Value Problems in the Presence of a Boundary Layer
... Remark 1. The results hold also without the restrictions imposed by (3), see [1], however the arguments become more complicated and the constants in the estimates will differ. Furthermore, note that when β > 0 then (3) can always be ensured for ε smaller than some threshold value ε 0 by a simple transformation u(x) =ũ(x)e χx with χ chosen appropriately. ...
... The results hold also without the restrictions imposed by (3), see [1], however the arguments become more complicated and the constants in the estimates will differ. Furthermore, note that when β > 0 then (3) can always be ensured for ε smaller than some threshold value ε 0 by a simple transformation u(x) =ũ(x)e χx with χ chosen appropriately. ...
... Bakhvalov meshes [3] are better adapted to the layers, but less simple in construction. A mesh generating function is defined by ...
Richardson extrapolation is applied to a simple first-order upwind difference scheme for the approximation of solutions of singularly perturbed convection-diffusion problems in one dimension. Robust α posteriori error bounds are derived for the proposed method on arbitrary meshes. It is shown that the resulting error estimator can be used to stear an adaptive mesh algorithm that generates meshes resolving layers and singularities. Numerical results are presented that illustrate the theoretical findings.
AMS subject classification (2020): 65L11, 65L50, 65L70
... Later, an improved order of convergence (from first-order to secondorder) was established by Linß and Madden in [18] concerned with same equation as in [21]. In [19], Linß and Madden considered a system of m ≥ 2 equations assuming the coupling matrix a strongly diagonally dominant with the conditions 1] a ik a ii < 1, i = 1, 2, . . . , m. ...
... In their work, they compared their parameter uniform results using Shishkin, Bakhvalov, and equidistribution meshes. Taking k = , ∀k and a coercive coupling matrix A A A, Bakhvalov [1] established second-order convergence in his paper. Das and Natesan [7] dealt with a system of BVPs with Robin-type boundary conditions using a hybrid scheme that combines central difference and cubic spline approximations in different regions. ...
... Under these assumptions, we establish the following maximum principle for the differential operator L L L and prove that it satisfies the stability result in the maximum norm. Lemma 2.1 Assume that y y y ∈ (C (2,1) (Q) ∩ C (0,0) (Q)) m such that y y y(x, 0) ≥ 0 on (0, 1) and y y y(0, t) ≥ 0, y y y(1, t) ≥ 0 on (0, T ]. Then Ly Ly Ly ≥ 0, ∀ (x, t) ∈ Q implies that y y y ≥ 0, ∀ (x, t) ∈ Q. ...
This article presents a uniformly convergent numerical technique for a time-dependent reaction-dominated singularly perturbed system, including the same diffusion parameters multiplied with second-order spatial derivatives in all equations. Boundary layers are observed in the solution components for the small parameter. The proposed numerical technique consists of the Crank–Nicolson scheme in the temporal direction over a uniform mesh and quadratic B-splines collocation technique over an exponentially graded mesh in the spatial direction. We derived the robust error estimates to establish the optimal order of convergence. Numerical investigations confirm the theoretical determinations and the proposed method’s efficiency and accuracy.
... The rest of the paper is organized as follows. In Section 2, we give a priori information of the solution to (1), which the Bakhvalov-type mesh is based on, and define the finite element method. Some properties of mesh steps of the Bakhvalovtype mesh are also presented in this section. ...
... Let be any measurable subset of . We denote by and 1 the semi-norm in 1 and the norms in the Lebesgue spaces , respectively. When , we drop the subscript from the notation for simplicity. ...
... Bakhvalov's idea [1] is to map an equidistant grid by means of the boundary layer function like in Lemma 1. However, a nonlinear equation must be solved for Bakhvalov mesh (see [13,Part I §2.4.1]). ...
Supercloseness and postprocessing of the linear finite element method are studied on the Bakhvalov-type mesh for a singularly perturbed convection diffusion problem. Finite element analysis on this kind of mesh has always been an open problem. The difficulties arise from the width O(εln(1/ε)) of subdomain for the layer and nonuniformity of meshes in the layer. A novel interpolation is introduced to address difficulties from the width of subdomain for the layer. As a result, supercloseness of order two is obtained for the linear finite element method. Based on this supercloseness result, we propose and analyze a new postprocessing operator according to the mesh’s structure. Its stability is proved by means of numerical quadrature. Then, it is proved that the numerical solution after postprocessing converges second order. Numerical experiments verify these theoretical results.
... Note that there are different mesh techniques in the literature including different meshes for different variables [3,8,37]; different meshes for interior and boundary domains [3,34,35,38]. For boundary layers that are parallel to one of the axes, a Shishkin mesh, see for example, [10,13,14,33]; or a Bakhvalov' grid, see for example, [1,2], have been developed to capture the boundary layer effect. ...
... where κ is the curvature of the interface, γ is the surface tension parameter. Initially, we start with a star-shaped interface 1] domain. We set the surface tension coefficient as γ = 0.1. ...
Second order accurate Cartesian grid methods have been well developed for interface problems in the literature. However, it is challenging to develop third or higher order accurate methods for problems with curved interfaces and internal boundaries. In this paper, alternative approaches are developed for some interface and internal layer problems based on non-matching grids, in which two mesh sizes are used to obtain comparable high order accuracy near and away from an interface or an internal layer. For one-dimensional, or two-dimensional problems with straight interfaces or boundary layers that are parallel to one of the axes, the discussion is relatively easy. One of the challenges is how to construct a fourth order compact finite difference scheme at border grid points that connect two meshes. The idea is to employ a second order discretization near the interface in the fine mesh and a fourth order discretization away from the interface in the coarse and border grid points. For two-dimensional problems with a curved interface or an internal layer, a level set representation is utilized for which we can build a fine mesh within a tube |φ(x)|≤δh of the interface. A new super-third seven-point discretization that can guarantee the discrete maximum principle has been developed at hanging nodes. The coefficient matrices of the finite difference equations developed in this paper are M-matrices, which leads to the convergence of the finite difference schemes. Non-trivial numerical examples have confirmed the desired accuracy and convergence of the proposed method.
... Traditional numerical methods often fail to accurately capture these changes, which can result in errors across the entire domain. To address this issue, various methods such as Bakhavalov and Gartland meshes have been developed [22][23][24]. In this study, we analyze a standard finite element method combined with the Shishkin mesh, which is a type of local refinement strategy introduced by a Russian mathematician Grigorii Ivanovich Shishkin in 1988 [25]. ...
... Step 3: Then find u n ∈ V k n in the weak formulation of the equation (10) on the Shishkin mesh X N s for sufficiently large N (an even positive integer) independent of , The discretized linear systems corresponding to the stiffness matrix S, convection matrix C, and mass matrix M are obtained as shown below. Equation (25) represents the discretized linear system of (22), while equation (26) corresponds to the discretized linear system of (24). ...
This paper introduces an approach to decoupling singularly perturbed boundary value problems for fourth-order ordinary differential equations that feature a small positive parameter multiplying the highest derivative. We specifically examine Lidstone boundary conditions and demonstrate how to break down fourth-order differential equations into a system of second-order problems, with one lacking the parameter and the other featuring multiplying the highest derivative. To solve this system, we propose a mixed finite element algorithm and incorporate the Shishkin mesh scheme to capture the solution near boundary layers. Our solver is both direct and of high accuracy, with computation time that scales linearly with the number of grid points. We present numerical results to validate the theoretical results and the accuracy of our method.
... These assumptions ensure (see [14, Example III. 1.16]) that (1) has a unique solution in 1 0 2 for each 2 . Furthermore, the presence of the small parameter usually gives rises to an exponential layer of width ln 1 at the outflow boundary 0 and two parabolic layers of width ln 1 at the boundaries 0 and 1. ...
... There are roughly two types of layer-adapted grids: Bakhvalov-type meshes (B-type meshes) and Shishkin-type meshes (S-type meshes) (see [11]), which are approximations of Bakhvalov mesh [2] and generalizations of Shishkin mesh [18], respectively. The width of the mesh subdomain used to resolve the layer in problem (1) is ln 1 in the case of B-type meshes and ln in the case of S-type meshes, respectively. ...
This paper is to analyze a finite element method of any order on a Bakhvalov-type mesh in the case of 2D. By introducing a new interpolation according to the characteristics of layers, we show that the finite element method has uniform convergence of the optimal order with respect to the singular perturbation parameter. The result partially resolves an open problem introduced by Roos and Stynes (Comput. Methods Appl. Math. 15(4):531–550, 2015).
... acteristic of their solutions is the presence of layers. To fully resolve layers and obtain uniform convergence with respect to singular perturbation parameters, layer-adapted meshes have been introduced since 1960s [13] and have been an active research field [6,16]. Among them, the Shishkin meshes have been widely used and analyzed [9,14,17], because it has a very simple structure. ...
... Set χ = ξ in (13). According to the integration by parts, we derive ...
For singularly perturbed reaction-diffusion problems in 1D and 2D, we study a local discontinuous Galerkin (LDG) method on a Shishkin mesh. In these cases, the standard energy norm is too weak to capture adequately the behavior of the boundary layers that appear in the solutions. To deal with this deficiency, we introduce a balanced norm stronger than the energy norm. In order to achieve optimal convergence under the balanced norm in one-dimensional case, we design novel numerical fluxes and propose a special interpolation that consists of a Gauss-Radau projection and a local projection. Moreover, we generalize the numerical fluxes and interpolation, and extend convergence analysis of optimal order from 1D to 2D. Finally, numerical experiments are presented to confirm the theoretical results.
... In this manuscript, we analyze uniform supercloseness in the associated energy norm of a WG method on a Bakhvalov-type mesh [6] for a singularly perturbed twopoint boundary value problem. Bakhvalov-type meshes are popular layer-adapted meshes, which usually have better convergence than Shishkin-type meshes [23]. ...
... Bakhvalov mesh [6] was designed according to the structure of the layer function. This mesh is dense and graded in the layer region and produces better convergence than Shishkin mesh [16]. ...
In this paper, we analyze supercloseness in an energy norm of a weak Galerkin (WG) method on a Bakhvalov-type mesh for a singularly perturbed two-point boundary value problem. For this aim, a special approximation is designed according to the specific structures of the mesh, the WG finite element space and the WG scheme. More specifically, in the interior of each element, the approximation consists of a Gauß–Lobatto interpolant inside the layer and a Gauß–Radau projection outside the layer. On the boundary of each element, the approximation equals the true solution. Besides, with the help of over-penalization technique inside the layer, we prove uniform supercloseness of order k + 1 for the WG method. Numerical experiments verify the supercloseness result and test the influence of different penalization parameters inside the layer.
... e problem of nonsmoothness in the solution was firstly overcome by utilizing layer adaptive mesh methodologies. Bakhvalov [2] was the first who proposed special type of meshes, known as Bakhvalov meshes, to capture the boundary layers found in reactiondiffusion problems' solutions, and later on, these meshes were used and modified by Gartland [3] and others for convection-diffusion problems. Later on, Shishkin [4] introduced another special type of meshes, called Shishkin meshes, to generate parameter-uniform numerical schemes under the finite-difference framework. ...
... where ϕ i s are the test functions generated by moving least square approach as discussed in Section 2. e last terms in equation (45) are introduced due to the method of Lagrange multipliers to enforce the essential boundary conditions (2). Substituting the EFG approximation of field function v(x) and simplifying equation (45), we obtain ...
As it is well recognized that conventional numerical schemes are inefficient in approximating the solutions of the singularly perturbed problems (SPP) in the boundary layer region, in the present work, an effort has been made to propose a robust and efficient numerical approach known as element-free Galerkin (EFG) technique to capture these solutions with a high precision of accuracy. Since a lot of weight functions exist in the literature which plays a crucial role in the moving least square (MLS) approximations for generating the shape functions and hence affect the accuracy of the numerical solution, in the present work, due emphasis has been given to propose a robust weight function for the element-free Galerkin scheme for SPP. The key feature of nonrequirement of elements or node connectivity of the EFG method has also been utilized by proposing a way to generate nonuniformly distributed nodes. In order to verify the computational consistency and robustness of the proposed scheme, a variety of linear and nonlinear numerical examples have been considered and L∞ errors have been presented. Comparison of the EFG solutions with those available in the literature depicts the superiority of the proposed scheme.
... Various meshes have been proposed in the literature. The most frequently analysed are the exponentially graded mesh of Bakhvalov [1] and piecewise uniform mesh of Shishkin [12]. ...
... Using Bakhvalov-type meshes the analysis of one-dimensional boundary value problems becomes substantially more dicult, as is the analysis of the dicretization of higher dimensional boundary value problems. From the identity (7.4) a n b n = (a b)(a n 1 + a n 2 b + ::: + ab n 2 + b n 1 ); n 2 N; and h i 1 Now, collecting (3:1); (7:1); (7:2); (7:3) and (7:6), the statement of the lemma is therefore proven. Using (3:1); (7:10); (7:12); (7:13); (7:15); (7:16); (7:18) and (7:19) completes the proof of the lemma. ...
In this work we consider the singularly perturbed one-dimensional semi-linear reaction-diffusion problem " y (x) = f (x; y); x 2 (0; 1) ; y(0) = 0; y(1) = 0; where f is a nonlinear function. Here the second-order derivative is multiplied by a small positive parameter and consequently, the solution of the problem has boundary layers. A new difference scheme is constructed on a modified Shishkin mesh with O(N) points for this problem. We prove existence and uniqueness of a discrete solution on such a mesh and show that it is accurate to the order of N^{-2} ln^{2} N in the discrete maximum norm. We present numerical results that verify this rate of convergence.
... The first Bakhvalov mesh is demonstrated in [41], where the mesh-generating function belongs to the space 1 . However, ensuring smoothness requires solving a nonlinear equation, which can be computationally expensive; see [5]. ...
This study analyzes a high‐order weak Galerkin finite element method (WG‐FEM) to solve the one‐dimensional unsteady convection–diffusion equation with a nonlinear reaction term on a non‐uniform mesh. The proposed method employs piecewise polynomials of degree k≥1 in the interior, combined with a constant polynomial approximation at each element boundary. We apply WG‐FEM on a Bakhvalov mesh for spatial discretization, coupled with the Crank–Nicolson scheme for time discretization on a uniform mesh. The standard Lagrange interpolation is commonly used in error analysis for the finite element method when solving convection‐dominated problems. However, when applied to Bakhvalov‐type meshes, this interpolation can lead to instability, particularly within specific section of the mesh known as problematic region. To address this issue and accurately capture the solution's behavior in this region, a special interpolation, represented as ℜ, is introduced. The primary outcome of this study demonstrates that the weak Galerkin solution on the Bakhvalov mesh achieves a parameter‐free error bound of order 𝒪(N−k) in the spatial direction, along with second‐order convergence in the temporal direction. Numerous numerical experiments have been conducted to validate these theoretical findings.
... Due to the steep boundary layer, stable and accurate numerical solutions of such equations are hard to obtain. Special numerical techniques have been developed to resolve the layers [1,5,16,12]. Adaptive meshes are commonly used to generate meshes that have high resolution near the boundary layer [12]. ...
Singularly perturbed problems are known to have solutions with steep boundary layers that are hard to resolve numerically. Traditional numerical methods, such as Finite Difference Methods (FDMs), require a refined mesh to obtain stable and accurate solutions. As Physics-Informed Neural Networks (PINNs) have been shown to successfully approximate solutions to differential equations from various fields, it is natural to examine their performance on singularly perturbed problems. The convection-diffusion equation is a representative example of such a class of problems, and we consider the use of PINNs to produce numerical solutions of this equation. We study two ways to use PINNS: as a method for correcting oscillatory discrete solutions obtained using FDMs, and as a method for modifying reduced solutions of unperturbed problems. For both methods, we also examine the use of input transformation to enhance accuracy, and we explain the behavior of input transformations analytically, with the help of neural tangent kernels.
... Several branches of solution strategies have been developed. One is based on mesh refinement or grading toward layers [3,31,54,55]. The more popular alternative, in particular in engineering communities, is the class of stabilized methods. ...
This paper presents a novel multi-scale method for convection-dominated diffusion problems in the regime of large Péclet numbers. The method involves applying the solution operator to piecewise constant right-hand sides on an arbitrary coarse mesh, which defines a finite-dimensional coarse ansatz space with favorable approximation properties. For some relevant error measures, including the L 2 -norm, the Galerkin projection onto this generalized finite element space even yields ε -independent error bounds, ε being the singular perturbation parameter. By constructing an approximate local basis, the approach becomes a novel multi-scale method in the spirit of the Super-Localized Orthogonal Decomposition (SLOD). The error caused by basis localization can be estimated in an a posteriori way. In contrast to existing multi-scale methods, numerical experiments indicate ε -robust convergence without pre-asymptotic effects even in the under-resolved regime of large mesh Péclet numbers.
... In literature, various layer-adapted meshes, such as Bakhvalov mesh, Shishkin mesh, generalized Shishkin mesh, Bakhvalov-Shishkin mesh, etc., are constructed [16]. Bakhvalov [3] was the first one to introduce a fitted mesh in 1969. The mesh was generated with the help of a suitable mesh generating function. ...
We are focused on the numerical treatment of a singularly perturbed degenerate parabolic convection–diffusion problem that exhibits a parabolic boundary layer. The discretization and analysis of the problem are done in two steps. In the first step, we discretize in time and prove its uniform convergence using an auxiliary problem. In the second step, we discretize in space using an upwind scheme on a Bakhvalov-type mesh and prove its uniform convergence using the truncation error and barrier function approach, wherein several bounds derived for the mesh step sizes are used. Numerical results for a couple of examples are presented to support the theoretical bounds derived in the paper.
... Let σ > 0 be a user chosen mesh parameter. A mesh transition point τ is defined by [3] are better adapted to the layers, but less simple in construction. A mesh-generating function is defined by ...
Richardson extrapolation is applied to a simple first-order upwind difference scheme for the approximation of solutions of singularly perturbed convection-diffusion problems in one dimension. Robust a posteriori error bounds are derived for the proposed method on arbitrary meshes. It is shown that the resulting error estimator can be used to steer an adaptive mesh algorithm that generates meshes resolving layers and singularities. Numerical results are presented that illustrate the theoretical findings.
... To estimate the derivative bounds on and , we use the following result [44]. ...
... Commonly used layer-adapted meshes for solving singularly perturbed problems include Bakhvalovtype meshes and Shishkin-type meshes. Bakhvalov mesh is proposed for the first time in [2], its application needs a nonlinear equation which cannot be solved explicitly. In order to avoid this difficulty, meshes that arise from an approximation of Bakhvalov's mesh generating function are called Bakhvalov-type meshes, which are one of the most popular layer-adapted meshes, see details in [12]. ...
In this paper, we propose a weak Galerkin finite element method (WG) for solving singularly perturbed convection-diffusion problems on a Bakhvalov-type mesh in 2D. Our method is flexible and allows the use of discontinuous approximation functions on the meshe. An error estimate is devised in a suitable norm and the optimal convergence order is obtained. Finally, numerical experiments are given to support the theory and to show the efficiency of the proposed method.
... Our aim in this paper is to present a uniform numerical method for solving singularly perturbed nonlinear integro-differential equations and compare the obtained results on Bakhvalov and Shishkin type meshes. In addition to the subject, Bakhvalov mesh was developed by N. S. Bakhvalov in 1969 [7] and G. I. Shishkin designed the piecewise equidistant mesh [30]. ...
This article deals with the singularly perturbed nonlinear Volterra-Fredholm integro-differential equations. Firstly, some priori bounds are presented. Then, the finite difference scheme is constructed on non-uniform mesh by using interpolating quadrature rules [5] and composite numerical integration formulas. The error estimates are derived in the discrete maximum norm. Finally, theoretical results are performed on two examples and they are compared for both Bakhvalov (B-type) and Shishkin (S-type) meshes.
... These problems are encountered in science, economics, sociology, engineering, medical science, fluids mechanics, aerodynamics, magnetic dynamics, emission theory, reaction diffusion, light emitting waves, communication lines, plasma dynamics, purified gas dynamics, motion of mass, plastics, chemical reactor theory, seismology, oceanography, meteorology, electric current, ion acoustic waves and some physical modeling [16]- [21]. Also, Bakhvalov used a special transformation in the numerical solution of boundary layer problems [22]. Bitsadze and Samarskii obtained some generalizations of linear elliptic boundary value problems [23]. ...
In this study, we obtain approximate solution for singularly perturbed problem of differential equation having two integral boundary conditions. With this purpose, we propose a new finite difference scheme. First, we construct this exponentially difference scheme on a uniform mesh using the finite difference method. We use the quasilinearization method and the interpolating quadrature formulas to establish the numerical scheme. Then, as a result of the error analysis, we show that the method under study is convergent in the first order. Consequently, theoretical findings are supported by numerical results obtained with an example. Approximate solutions curves are compared on the chart to provide concrete indication. The maximum errors and convergence rates obtained are given on the table for different varepsilon and N values.
... Traditional numerical methods often fail to accurately capture these changes, which can result in errors across the entire domain. To address this issue, various methods such as Bakhavalov [11] and Gartland meshes [12] have been developed. In this study, we analyze a standard upwind finite difference method combined with the Shishkin mesh, which is a type of local refinement strategy introduced by a Russian mathematician Grigorii Ivanovich Shishkin in 1988 [13], [3]. ...
This paper introduces a numerical approach to solve singularly perturbed convection diffusion boundary value problems for second-order ordinary differential equations that feature a small positive parameter {\epsilon} multiplying the highest derivative. We specifically examine Dirichlet boundary conditions. To solve this differential equation, we propose an upwind finite difference method and incorporate the Shishkin mesh scheme to capture the solution near boundary layers. Our solver is both direct and of high accuracy, with computation time that scales linearly with the number of grid points. MATLAB code of the numerical recipe is made publicly available. We present numerical results to validate the theoretical results and assess the accuracy of our method. The tables and graphs included in this paper demonstrate the numerical outcomes, which indicate that our proposed method offers a highly accurate approximation of the exact solution.
... Bakhvalov mesh was first proposed in [2] to improve the convergence order. Bakhvalov-type mesh, as an approximation of Bakhvalov mesh, has been widely used to avoid solving nonlinear equations appearing on Bakhvalov mesh [10]. ...
On Bakhvalov-type mesh, uniform convergence analysis of finite element method for a 2-D singularly perturbed convection-diffusion problem with exponential layers is still an open problem. Previous attempts have been unsuccessful. The primary challenges are the width of the mesh subdomain in the layer adjacent to the transition point, the restriction of the Dirichlet boundary condition, and the structure of exponential layers. To address these challenges, a novel analysis technique is introduced for the first time, which takes full advantage of the characteristics of interpolation and the connection between the smooth function and the layer function on the boundary. Utilizing this technique in conjunction with a new interpolation featuring a simple structure, uniform convergence of optimal order k+1 under an energy norm can be proven for finite element method of any order k. Numerical experiments confirm our theoretical results.
... Starting in the late 1960s, in this evolution process, several numerical methods (independent of ε) have been constructed for a scalar reaction-diffusion equation (see, [2,38,43,45] and the references therein). On the other hand, less effort has been devoted to systems of reaction-diffusion boundary value problems. ...
A parameter-uniform numerical scheme for a system of weakly coupled singularly perturbed reaction-diffusion equations of arbitrary size with appropriate boundary conditions is investigated. More precisely, quadratic B-spline basis functions with an exponentially graded mesh are used to solve a × system whose solution exhibits parabolic (or exponential) boundary layers at both endpoints of the domain. A suitable mesh-generating function is used to generate the exponentially graded mesh. The decomposition of the solution into regular and singular components is obtained to provide error estimates. A convergence analysis is addressed, which shows a uniform convergence of the second order. To validate the theoretical findings, two test problems are solved numerically.
... Bakhvalov mesh is originally introduced and constructed for the layer functions in SPPs in [2]. The mesh points of the Bakhvalov mesh are given in terms of a piecewise C 1 continuous mesh generating function. ...
In this paper, we propose a weak Galerkin finite element method (WG-FEM) for solving two-point boundary value problems of convection-dominated type on a Bakhvalov-type mesh. A special interpolation operator which has a simple representation and can be easily extended to higher dimensions is introduced for convection-dominated problems. A robust optimal order of uniform convergence has been proved in the energy norm with this special interpolation using piecewise polynomials of degree on interior of the elements and piecewise constant on the boundary of each element. The proposed finite element scheme is independent of parameter and since the interior degree of freedom can be eliminated efficiently from the resulting discrete system, number of unknowns of the proposed method is comparable with the standard finite element methods. An optimal order of uniform convergence is derived on Bakhvalov-type mesh. Finally, numerical experiments are given to support the theoretical findings and show the efficiency of the proposed method.
... The purpose of this work is to formulate and analyze an ε−uniformly convergent numerical method for solving the problem (1.1) . The novelty of the presented method, unlike Shishkin mesh developed by [30] and Bakhvalov mesh developed by [2] , does not require a priori information about the location and width of the boundary layer. The methods developed by [7,[12][13][14]23] for Eq. ...
This paper deals with the numerical treatment of a singularly perturbed unsteady non-linear Burger-Huxley problem. Due to the simultaneous presence of a singular perturbation parameter and non-linearity in the problem applying classical numerical methods to solve this problem on a uniform mesh are unable to provide oscillation-free results unless they are applied with very fine meshes inside the region. Thus, to resolve this issue, a uniformly convergent computational scheme is proposed. The scheme is formulated:
•First, the non-linear singularly perturbed problem is linearized using the Newton-Raphson-Kantorovich quasilinearization technique.
•The resulting linear singularly perturbed problem is semi-discretized in time using the implicit Euler method to yield a system of singularly perturbed ordinary differential equations in space.
•Finally, the system of singularly perturbed ordinary differential equations are solved using fitted exponential cubic spline method.
The stability and uniform convergence of the proposed scheme are investigated. The scheme is stable and ε−uniformly convergent with first order in time and second order in space directions. To validate the applicability of the proposed scheme several test examples are considered. The obtained numerical results depict that the proposed scheme provides more accurate results than some methods available in the literature.
... Bakhvalov mesh first appeared in [21] and is graded in the layer. Its applications require the solution of a nonlinear equation. ...
A finite element method of any order is applied on a Bakhvalov‐type mesh to solve a singularly perturbed convection–diffusion equation in 2D, whose solution exhibits exponential boundary layers. A uniform convergence of (almost) optimal order is proved by means of a carefully defined interpolant.
... Significant research efforts have focused on devising parameter-robust numerical methods, i.e., methods with a uniform convergence rate with respect to the singular perturbation parameter. Such methods have been commonly designed by using layer-adapted meshes, such as the Bakhvalov mesh [5], the Shishkin mesh [35], and the Spectral Boundary Layer mesh in the context of the p/hp-version FEM [26,27], and uniform convergence rates were typically obtained for problems with sufficiently smooth data. In recent years, parameter robust methods with error estimates in a so-called balanced norm have attracted considerable research interest [21,28,32]. ...
A generalized finite element method is proposed for solving a heterogeneous reaction-diffusion equation with a singular perturbation parameter , based on locally approximating the solution on each subdomain by solution of a local reaction-diffusion equation and eigenfunctions of a local eigenproblem. These local problems are posed on some domains slightly larger than the subdomains with oversampling size . The method is formulated at the continuous level as a direct discretization of the continuous problem and at the discrete level as a coarse-space approximation for its standard FE discretizations. Exponential decay rates for local approximation errors with respect to and (at the discrete level with h denoting the fine FE mesh size) and with the local degrees of freedom are established. In particular, it is shown that the method at the continuous level converges uniformly with respect to in the standard norm, and that if the oversampling size is relatively large with respect to and h (at the discrete level), the solutions of the local reaction-diffusion equations provide good local approximations for the solution and thus the local eigenfunctions are not needed. Numerical results are provided to verify the theoretical results.
... At the point of discontinuity, we used a three-point scheme to resolve it. In the Shishkin-Bakhvalov mesh, we choose transition point as in Shishkin mesh and use graded mesh (as in Bakhvalov [1]) in the layer region. In the outer region a uniform mesh is used. ...
In this article, we have considered a time-dependent two-parameter singularly perturbed parabolic problem with discontinuous convection coefficient and source term. The problem contains the parameters and multiplying the diffusion and convection coefficients, respectively. A boundary layer develops on both sides of the boundaries as a result of these parameters. An interior layer forms near the point of discontinuity due to the discontinuity in the convection and source term. The width of the interior and boundary layers depends on the ratio of the perturbation parameters. We discuss the problem for ratio . We used an upwind finite difference approach on a Shishkin-Bakhvalov mesh in the space and the Crank-Nicolson method in time on uniform mesh. At the point of discontinuity, a three-point formula was used. This method is uniformly convergent with second order in time and first order in space. Shishkin-Bakhvalov mesh provides first-order convergence; unlike the Shishkin mesh, where a logarithmic factor deteriorates the order of convergence. Some test examples are given to validate the results presented.
... Starting in the late 1960s, in this evolution process, several numerical methods (independent of ε) have been constructed for a scalar reaction-diffusion equation (see, [2,38,43,45] and the references therein). On the other hand, less effort has been devoted to systems of reaction-diffusion boundary value problems. ...
A parameter-uniform numerical scheme for a system of weakly coupled singularly perturbed reaction-diffusion equations of arbitrary size with appropriate boundary conditions is investigated. More precisely, quadratic B-spline basis functions with an exponentially graded mesh are used to solve a system whose solution exhibits parabolic (or exponential) boundary layers at both endpoints of the domain. A suitable mesh generating function is used to generate the exponentially graded mesh. The decomposition of the solution into regular and singular components is obtained to provide error estimates. A convergence analysis is addressed, which shows a uniform convergence of the second order. To validate the theoretical findings, two test problems are solved numerically.
... Initially, Bakhvalov introduced a mesh based on a generic function to tackle such kind of boundary layers (see [8]). Vulanovic in 1983, proposed simplest Bakhvalov mesh named B-mesh by replacing the term − ln([q −t]/q) with t/(t −q) in the mesh generating function (see [24,27]). ...
This article deals with the class of singularly perturbed convection-diffusion problem with time delay. A parameter uniform numerical method is developed, and its detailed analysis is done. To discretize the spatial domain, a harmonic mesh H(ℓ) is used, which gives more accurate results in comparison with Shishkin, S(ℓ) and Bakhvalov mesh. Numerical experiments are carried out to validate the proposed method. The computational results on H(ℓ) mesh have been compared with the other existing meshes like B-mesh, Shishkin mesh and S(ℓ) mesh.
... One strategy is to use the location and width of the layers. Some well-known meshes developed using this strategy are Shishkin mesh [3], Bakhvalov mesh [24], generalized Shishkin mesh [25], and so on. The other strategy is to use the entropy of the solution [26] or the mesh partial differential equations [27]. ...
The purpose of this paper is to introduce a high order convergent numerical method for singularly perturbed time dependent problems using mesh equidistribution. The discretization is based on the backward Euler scheme in time and a high order non-monotone scheme in space. In time direction we consider a uniform mesh, while in spatial direction we construct an adaptive mesh through equidistribution of a monitor function involving appropriate power of the solution’s second derivative. The method is analysed in two steps, splitting the time and space discretization errors. We establish that the method is uniformly convergent with optimal order having order one in time and order four in space. Further, we use the Richardson extrapolation technique for improving the order of convergence from one to two in time. Numerical experiments are presented to confirm the theoretically proven convergence result.
... This is especially the case for the very small boundary layers of acoustic waves. Finite difference schemes or finite element meshes specially adapted close to walls have been proposed for various model problems with boundary layers [8][9][10][11] which regain the optimal convergence rate of the numerical schemes; see also the review papers. 12,13 With impedance boundary conditions, the boundary layers need not to be resolved at all as they are posed for the macroscopic part of the solution. ...
We present impedance boundary conditions for the viscoacoustic equations for approximative models that are in terms of the acoustic pressure or in terms of the macroscropic acoustic velocity. The approximative models are derived by the method of multiple scales up to order 2 in the boundary layer thickness. The boundary conditions are stable and asymptotically exact, which is justified by a complete mathematical analysis. The models can be discretized by finite element methods without resolving boundary layers. In difference to an approximation by asymptotic expansion for which for each order 1 PDE system has to be solved, the proposed approximative are solutions to one PDE system only. The impedance boundary conditions for the pressure of first and second orders are of Wentzell type and include a second tangential derivative of the pressure proportional to the square root of the viscosity and take thereby absorption inside the viscosity boundary layer of the underlying velocity into account. The conditions of second order incorporate with curvature the geometrical properties of the wall. The velocity approximations are described by Helmholtz‐like equations for the velocity, where the Laplace operator is replaced by ∇div, and the local boundary conditions relate the normal velocity component to its divergence. The velocity approximations are for the so‐called far field and do not exhibit a boundary layer. Including a boundary corrector, the so‐called near field, the velocity approximation is accurate even up to the domain boundary. The results of numerical experiments illustrate the theoretical foundations.
... The Shishkin mesh has been successfully used to solve singular convection-diffusion problems and is much simpler than other graded meshes such as the Bakhvalov mesh [8]. ...
A thin layer will develop at the boundary if the incoming angular flux is anisotropic in thick diffusive neutron transport problems. Solving such singularly perturbed problems, which have non-smooth solutions with singularity near the boundary, is computationally challenging. Standard finite difference schemes on a uniform mesh cannot yield-uniform convergence, where is a small parameter, while it can be achieved on a suitable piecewise-uniform Shishkin mesh. We present a formal error analysis of the diamond difference (DD) method and step difference (SD) method for solving the SN neutron transport equation. The analysis can be extended to other finite difference methods. Numerical results are presented to confirm the error estimates and the advantages of the Shishkin mesh.
... To overcome this difficulty for such problems, various fitted mesh (a prior refined mesh) methods are constructed. Popular fitted meshes are Bakhvalov mesh (Bakhvalov 1969), Vulanovic mesh (Vulanović 1987), Shishkin mesh (Shishkin 1988), graded mesh (Gartland 1988). Among all meshes, piecewise uniform Shishkin mesh is most commonly preferred due to simplicity, but on this mesh the order of convergence is deteri- orated by the presence of ln N term where N is the number of sub-intervals. ...
In this article, we construct a numerical method to solve a two-parameter singularly perturbed problem in two dimensions on a tensor product mesh of two exponentially graded mesh. The finite difference scheme is proved to be second-order uniformly convergent when μ2/ε→0 as ε→0. Numerical experiments validate the obtained error estimates.
... and that Ω is a smooth domain meaning that ∂Ω is an analytic curve. We mention that problem (1), (2) has been studied in [5] (see also [4]) with ε 2 a fixed constant, hence only one singular perturbation parameter. Here we will focus on the case ε 1 < ε 2 , and in particular we assume ...
We consider fourth order singularly perturbed boundary value problems with two small parameters, and the approximation of their solution by the version of the finite element method on the spectral boundary layer mesh from Melenk et al. We use a mixed formulation requiring only basis functions in two‐dimensional smooth domains. Under the assumption of analytic data, we show that the method converges uniformly, with respect to both singular perturbation parameters, at an exponential rate when the error is measured in the energy norm. Our theoretical findings are illustrated through numerical examples, including results using a stronger (balanced) norm.
... Compared to the quasi-uniform mesh, this kind of meshes can capture the change of layers better. Among those the most representative ones are Shishkin mesh [9] and Bakhvalov mesh [2], and numerical experiments show the superiority of Bakhvalov mesh. ...
On a Bakhvalov-type mesh widely used for boundary layers, we consider the finite element method for singularly perturbed elliptic problems with two parameters on the unit square. It is a very challenging task to analyze uniform convergence of finite element method on this mesh in 2D. The existing analysis tool, quasi-interpolation, is only applicable to one-dimensional case because of the complexity of Bakhvalov-type mesh in 2D. In this paper, a powerful tool, Lagrange-type interpolation, is proposed, which is simple and effective and can be used in both 1D and 2D. The application of this interpolation in 2D must be handled carefully. Some boundary correction terms must be introduced to maintain the homogeneous Dirichlet boundary condition. These correction terms are difficult to be handled because the traditional analysis do not work for them. To overcome this difficulty, we derive a delicate estimation of the width of some mesh. Moreover, we adopt different analysis strategies for different layers. Finally, we prove uniform convergence of optimal order. Numerical results verify the theoretical analysis.
... Since we cannot use standard ideas, we take the approach of rewriting (1) as a coupled system of real-valued problems, and establish that the coefficient matrix for this system is positive definite. From that we conduct the analysis using ideas due to Bakhvalov [1], as illustrated by Kellogg et al. [2]. We also present numerical results that demonstrate that our theoretical error estimates are sharp. ...
Objective . The aim of the study is to solve a problem aimed at assessing the characteristics of electromagnetic wave scattering on hollow structures whose dimensions belong to the resonant region.
Method . The dimensions of the hollow structures with maximum scattering characteristics were determined by a combination of the integral equation method and the optimization method. Scattering at the edges of the aperture of the hollow structure is taken into account. It is proposed to use the Mathieu equation to determine the flow characteristics. The stages of the algorithm for calculating the Mathieu functions are given, which were used during the implementation of the computer program. Parseval's equality is used for the integral transformation.
Result . A mathematical model and an algorithm for numerical analysis of the scattering features of plane radio waves on hollow structures that are components of complex-shaped objects, antenna-feeder lines, and antenna devices have been created. The results of test calculations have been obtained. The structure of the subsystem for analyzing complex-shaped hollow structures has been proposed.
Conclusion . A priori estimates are obtained for the solution of a boundary value problem in a strip of higher-order elliptic equations degenerating to a cubic equation in one variable. Conditions for achieving a priori estimates are shown; additional spaces are introduced for this purpose. The problem is studied in weighted spaces of the S.L. Sobolev type. Two theorems related to the boundary value problem in a strip for one class of degenerate elliptic equations of high order are considered, and an analysis of the possibilities of obtaining an a priori estimate is carried out. Weighted spaces give an a priori estimate for the solution of a boundary value problem in a strip for a higher-order elliptic equation degenerating to a cubic equation on one of the boundaries of the strip in one of the variables.
This article examines periodic Sobolev reports with a singular deviation, which causes significant difficulties in numerical approximation due to the presence of sharp or boundary layers. A stable quantitative method for the effective solution of such problems in the Bakhvalov lattice, a special grid for the deviant action of the solution, is proposed. Singularly perturbed periodic Sobolev problems create significant difficulties in numerical approximation due to the presence of sharp layers or boundary layers. Our proposed reliable numerical method for efficiently solving such problems on the Bakhvalov grid, a specialized grid, is designed to account for the singular behavior of the solution. First, an asymptotic analysis of the exact solution is performed. Then a finite difference scheme is created by applying quadrature interpolation rules to an adaptive network. The stability and convergence of the presented algorithm in a discrete maximum norm is analyzed. The results show that the proposed approach provides an accurate approximation of the solution for singular problems while maintaining computational efficiency.
We propose a new method for the construction of layer-adapted meshes for singularly perturbed differential equations (SPDEs), based on mesh partial differential equations (MPDEs) that incorporate a posteriori solution information. There are numerous studies on the development of parameter-robust numerical methods for SPDEs that depend on the layer-adapted mesh of Bakhvalov. In [R. Hill, N. Madden, Numer. Math. Theory Methods Appl. 14, 559–588], a novel MPDE-based approach for constructing a generalisation of these meshes was proposed. Like with most layer-adapted mesh methods, the algorithms in that article depended on detailed derivations of a priori bounds on the SPDE's solution and its derivatives. In this work, we extend that approach so that it instead uses a posteriori computed estimates of the solution. We present detailed algorithms for the efficient implementation of the method, and numerical results for the robust solution of two-parameter reaction-convection-diffusion problems, in one and two dimensions. We also provide full FEniCS code for a one-dimensional example.
In this paper, we propose a weak Galerkin finite element method (WG-FEM) for solving nonlinear boundary value problems of reaction–diffusion type on a Bakhvalov-type mesh. A robust optimal order of uniform convergence on a Bakhvalov-type mesh has been proved in the energy norm and a stronger balanced norm using piecewise polynomials of degree on interior of the elements and piecewise constant on the boundary of each element. The proposed finite element scheme is independent of parameter and since the interior degree of freedom can be eliminated efficiently from the resulting discrete system, number of unknowns of the proposed method is comparable with the standard finite element methods. For the first time, a uniform error estimate has been established in the energy norm and in a balanced norm using higher order polynomials on a Bakhvalov-type mesh. Finally, numerical experiments are given to support the theoretical findings and show the efficiency of the proposed method.
We consider a singularly perturbed initial-third boundary value Sobolev problems. Firstly, the asymptotic behaviour of the exact solution is analysed. Then, a second-order finite difference scheme is constructed on the special non-uniform mesh. By using energy estimate, the stability and convergence of the proposed scheme are investigated in the discrete maximum norm. Finally, three numerical examples are solved to validate the theory.
In the present paper, Robin boundary value problem for a system of singularly perturbed reaction-diffusion equations with discontinuous source term is studied. The highest order derivative in each equation is multiplied by the perturbation parameters which are different in magnitude. The considered system does not obey maximum principle. Forward-backward approximation is used for the Robin boundary conditions and a central finite difference approximation is proposed for the differential system in conjunction with piecewise uniform Shishkin meshes and graded Bakhvalov meshes. The scheme is proved to be an almost first-order parameter uniform convergent. Numerical experiments are presented which are in line with the theoretical findings.
This paper presents a multi-scale method for convection-dominated diffusion problems in the regime of large P\'eclet numbers. The application of the solution operator to piecewise constant right-hand sides on some arbitrary coarse mesh defines a finite-dimensional coarse ansatz space with favorable approximation properties. For some relevant error measures, including the -norm, the Galerkin projection onto this generalized finite element space even yields -independent error bounds, being the singular perturbation parameter. By constructing an approximate local basis, the approach becomes a novel multi-scale method in the spirit of the Super-Localized Orthogonal Decomposition (SLOD). The error caused by basis localization can be estimated in an a-posteriori way. In contrast to existing multi-scale methods, numerical experiments indicate -independent convergence without preasymptotic effects even in the under-resolved regime of large mesh P\'eclet numbers.
In this article, we analyze convergence of a weak Galerkin method on Bakhvalov-type mesh. This method uses piecewise polynomials of degree k≥1 on the interior and piecewise constant on the boundary of each element. To obtain uniform convergence, we carefully define the penalty parameter and a new interpolant which is based on the characteristic of the Bakhvalov-type mesh. Then the method is proved to be convergent with optimal order, which is confirmed by numerical experiments.
We consider fourth-order singularly perturbed eigenvalue problems in one-dimension and the approximation of their solution by the h version of the Finite Element Method (FEM). In particular, we use a C 1 {C^{1}} -conforming FEM with piecewise polynomials of degree p ≥ 3 {p\geq 3} defined on an exponentially graded mesh. We show that the method converges uniformly, with respect to the singular perturbation parameter, at the optimal rate when the error in the eigenvalues is measured in absolute value and the error in the eigenvectors is measured in the energy norm. We also illustrate our theoretical findings through numerical computations for the case p = 3 {p=3} .
The main aim of this article is to present an a priori estimate of an exponential layer and thereby a new mesh (G-mesh) for a class of singularly perturbed problems with discontinuous data. We have also proved that the standard upwind scheme on the proposed mesh is parameter uniformly convergent and is of almost first order. To demonstrate the efficiency of the proposed mesh we carried out a few numerical experiments comparing the performance of the proposed G-mesh with other standard meshes available in the literature.
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