Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems
Chapters (8)
Before surveying a few of the most important ideas from the literature for constructing layer-adapted meshes, we shall introduce some basic concepts for describing layer-adapted meshes.
Throughout \(\varpi :0\; = \;x_0 < x_1 < ... < x_N = 1\) denotes a generic mesh with N subintervals on [0, 1], while ω is the set of inner mesh nodes. Set \(I_i : = \;[x_i - 1,x_i ].\). The local mesh sizes are \(h_i : = x_i - x_{i - 1} ,i = 1,...,N\), while the maximum step size is \(h: = \mathop {\max }\limits_{i = 1,...,N} hi.\).
In this chapter, we gather a number of analytical properties for singularly perturbed boundary-value problems for second-order ordinary differential equations of the general type $$ - \in u^ - bu^{'} + cu = f\text{ in }(0,1),u(0) = \gamma _0 ,\text{ }u(1) = \gamma _1 ,$$ with a small positive parameter ε and functions b, c, f : [0, 1] → IR, and of its vector-valued counterpart $$ - Eu^ - Bu^{'} + Au = f\text{ in }(0,1),u(0) = \gamma _0 ,\text{ }u(1) = \gamma _1 ,$$ with \( E = \text{diag}(\in ),\in = (\in _1 ,....,\in _\ell )^T \) and small positive constants \(\in _i ,i = 1,...,\ell ,\) with matrix-valued functions A, B : \( [0,1] \to IR^{\ell ,\ell } ,\), and vector-valued functions \( f,u:[0,1] \to IR^\ell \).
This chapter is concerned with finite-difference discretisations of the stationary linear convection-diffusion problem
$$Lu\,: = - \varepsilon u'' - bu' + cu = f\,\,{\rm in}\,(0,1),\;\;u(0) = \gamma _0 ,\;u(1) = \gamma _1 ,$$
with b ≥ β > 0 on [0, 1]. For the sake of simplicity we shall assume that
$$c \ge 0\,\,{\rm and}\;\;{\rm b'} \ge 0\;\;\;\;\;{\rm on[0,1]}{\rm .}$$
Using (4.1) as a model problem, a general convergence theory for certain firstand second-order upwinded difference schemes on arbitrary and on layer-adapted meshes is derived. The close relationship between the differential operator and its upwinded discretisations is highlighted.
In this chapter we consider finite element and finite volume discretisations of $$Lu\,: = - \varepsilon u'' - bu' + cu = f\,\,{\rm in}\,(0,1), \,\,\ u(0) = u (1) = 0,$$
with b ≥ β > 0. Its associated variational formulation is: Find \(u \in H_0^1 (0,1)\) such that
$$a(u, v) = f(v)\,\,\, {\rm for\, all}\,\, v \in H_0^1 (0,1),$$ (5.2)
where
$$a(u,v): = \;\varepsilon (u',v') - (bu',v) + (cu,v)$$
and
$$f(v):=(f,v):= \int_0^1 {(fv)(x)dx.}$$ (5.3)
Throughout assume that
$$c + b' / 2 \ge \gamma > 0.$$ (5.4)
This condition guaranties the coercivity of the bilinear form in (5.2):
$$\||v |\|_\varepsilon^2 := _\varepsilon \|v' \|_0^2 + \gamma \|v \|^2_0 \le a(u,v)\,\,\,\, {\rm for\, all}\,\,\,\, v \in H^1_0 (0, 1).$$
This is verified using standard arguments, see e.g. [141]. If b ≥ β > 0 then (5.4) can always be ensured by a transformation \(\bar u(x) = u(x)e^{\delta x}\) with δ chosen appropriately. We assume this transformation has been carried out.
This chapter is concerned with discretisations of the stationary linear reaction- 4
convection-diffusion problem $$ - \varepsilon _d u^ - \varepsilon _c bu + cu = f\text{ in (0,1), }u(0) = \gamma _0 ,u(1) = \gamma _1 ,$$ with b ≥ 1 and c ≥ 1 on [0, 1].In particular, we shall study the special case of scalar reaction-diffusion problems
$$ - \varepsilon _d u^ - \varepsilon _c bu + cu = f\text{ in (0,1), }u(0) = \gamma _0 ,u(1) = \gamma _1 ,$$
and its vector-valued counterpart
$$ - E^2 u'' + Au = f\;\;{\rm in}\;{\rm (0,1),}\;\;\;\;u(0) = \gamma _0 ,\;\;u(1) = \gamma _1 .
In this chapter numerical methods for singularly perturbed reaction-diffusion equations on the square are studied. Find \( u \in C^2 (\Omega ) \cup C(\bar \Omega ) \) such that $$ Lu: = - \varepsilon ^2 \Delta u + cu = f\text{ in }\Omega \text{ = (0,1)}^\text{2} ,\text{ }u|\partial \Omega = g
This chapter is devoted to numerical methods for the convection-diffusion problem $$- \varepsilon \Delta u - b\nabla u + cu = f\;in\;\Omega = (0,1)^2 ,\;u|_{\partial \Omega } = 0,$$ (9.1) with b
1 ≥ β1 > 0, b
2 ≥ β2 > 0 on [0,1]2, i.e., problems with regular boundary layers at the outflow boundary x = 0 and y = 0. The analytical behaviour of the solution of (9.1) was studied in Sect. 7.3.1.
Results for problems with characteristic layers will only be mentioned briefly.
... Linss analyzes them as finite-difference schemes applying the technique based on the discrete Green's function and a hybrid stability inequality, introduced in [14]. This result is repeated in Linss' book [6, Section 5.4, Theorem 5.27], published in 2010. It is puzzling that after the publication of Linss' result, the singular-perturbation researchers continue using the upwind scheme on the Shishkin mesh instead of switching to the Samarskii scheme or some other simple scheme from the same class, like the Runchal-Spalding scheme [15,16]. ...
... Numerical methods for singular perturbation problems have to produce approximate solutions that are accurate uniformly in . One class of such methods uses finite-difference schemes on special layer-adapted discretization meshes, such as the Bakhvalov [2,3] and Shishkin [4][5][6] meshes. In this paper, we only consider the original Shishkin mesh [4]. ...
... [18,Theorem 2.97]. The proofs in [6,17] use Linss' version [6,Theorem 3.48] of the Shishkin decomposition of the solution . ...
... wherê+ 1 ℎ and +1 ℎ are the solutions obtained from (30) and (31) ...
... Theorem 3.9. Let̂∈ +1 (Ω ) and ∈ +1 (Ω ) be the solutions of the problems (7) and (8), respectively, and̂+ 1 ℎ ( , ) and +1 ℎ ( , ) be the numerical solutions computed by (30) and (31) respectively. Let̂+ 1 ℎ, =̂−̂+ 1 ℎ ( , ) and +1 ℎ, = − +1 ℎ ( , ) be the corresponding errors. ...
... (69) Theorem 4.4. Let̂∈ +1 (Ω ) and ∈ +1 (Ω ) be the solutions of the problems (7) and (8), respectively, and̂+ 1 ℎ ( , ) and +1 ℎ ( , ) be the numerical solutions computed by (30) and (31) respectively. ...
In this article, we propose an Alternating Direction Implicit (ADI) type operator splitting weak Galerkin finite element method (WG‐FEM) for solving a parabolic singularly perturbed problem with two‐parameters in 2D over a layer‐adapted mesh. The suggested operator splitting approach divides the original model problem into two subproblems each in 1D, then solving each subproblem using WG‐FEM in spatial direction eventually reduces the computational difficulty and high storage requirements. Backward–Euler time discretization has been taken over a uniform mesh. Stability and convergence results have been proved for the fully‐discrete scheme. Numerical examples are presented corroborating in practice our theoretical findings.
... Due to the presence of boundary layers in the solution, layer-adapted meshes are often used in developing uniformly convergent numerical methods for singularly perturbed problems. 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. ...
... Some works on Shishkin and Bakhvalov-type meshes can be found in [9,10,12,13,26] and [1,2,15,25,28,30] respectively. To gain a more comprehensive understanding of the layer-adapted mesh design, we encourage readers to explore [8,16,19,27]. ...
... Proof We first prove (16). Since ...
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.
... Discontinuous coefficients introduce abrupt changes or jumps in the system's material properties or physical characteristics. Moreover, the problem being singularly perturbed, the solution of these equations exhibits a multiscale character since the corresponding degenerate system fails to satisfy the given boundary data [10,11]. There are narrow regions across which the solution changes rapidly and displays layer behavior. ...
... where 0 < ( 1 is the small perturbation parameter, d c ðxÞ ¼ dðx À cÞ with c 2 X is the Dirac-delta function, aðxÞ ! a 1 > 0 for x 2 ð0, cÞ, and aðxÞ ! a 2 > 0 for x 2 ðc, 1Þ. The solution exhibits an exponential boundary layer with a correction function e Àa1x= at x ¼ 0 and an interior layer e Àa2ðxÀcÞ= at the right of x ¼ c, attributed to concentrated sources or the discontinuity of the convective field [10]. Following the argument from Ref. [38], the problem (1) must be interpreted in a distributional context. ...
... The operator L satisfies the maximum principle [39], and the problem (1) has a unique solution, u 2 C½0, 1 \ C 2 ðð0, cÞ [ ðc, 1ÞÞ [10]. Moreover, from Ref. [38], we have ...
A posteriori error analysis of defect correction method for singular perturbation problems with discontinuous coefficient and point source The paper presents a defect correction method to solve singularly perturbed problems with discontinuous coefficient and point source. The method combines an inexpensive, lower-order stable, upwind difference scheme and a higher-order, less stable central difference scheme over a layer-adapted mesh. The mesh is designed so that most mesh points remain in the regions with rapid transitions. A posteriori error analysis is presented. The proposed numerical method is analysed for consistency, stability and convergence. The error estimates of the proposed numerical method satisfy parameter-uniform second-order convergence on the layer-adapted grid. The convergence obtained is optimal because it is free from any logarithmic term. The numerical analysis confirms the theoretical error analysis.
... The solution U(x, t k+1 ) of the semidiscretized problem (10) admit a decomposition into smooth and singular components [20]. We write ...
... Here, k 1 (ε, μ) and k 2 (ε, μ) are constants to be chosen to satisfy the jump conditions at x = 1 given in (19) and (20). Moreover, constants j 1 (ε, μ) and j 2 (ε, μ) are to be chosen separately for the cases p 0 μ 2 ≤ γ ε and p 0 μ 2 ≥ γ ε to satisfy the requirements for the bounds of the singular component. ...
... The exact solutions of the above examples are not known for comparison. Therefore, the double mesh principle [20] is used to estimate the maximum absolute error and rate of convergence. The maximum absolute error (E N, t ε ) and order of convergence (R N, t ε ) are Tables 1-4. ...
... The convergence rate is O((N -1 ln N) k + 1 ), where N is the number of total elements. In Ref. [6], the results were extended to the Bakhvalov mesh [7][8] , the interpolation error was proved to be O(N -(k + 1) ) and the integration error was O(N -k ). Furthermore, a new k point interpolation formula was discussed in Ref. [9]. ...
... From (7), (8) and (21), one gets ...
Traditional numerical integration requires sufficient smoothness of the integrand to achieve high-order algebraic accuracy. If the function has a boundary layer with large gradient, the composite integration formula on the uniform mesh will produce very large integration errors. In this paper, we study the Newton-Cotes formula based on Lagrange interpolation functions, local L2 projection approximating the integrand, and Gauss integration. On the Shishkin mesh, we establish an optimal-order integration error estimate uniformly in the perturbation parameter. The convergence rate is the same as that for the smooth function. Numerical experiments confirm the sharpness of our theoretical results.
... Over the years, various numerical methods have been developed to handle SPPs, such as FEMs or finite difference (FD) methods. However, these methods yield unsatisfactory and irrelevant outcomes due to boundary layers unless the parameter is only slightly more significant than the mesh size [9]. One efficient way to handle singularly perturbed problems is to use layer-adapted meshes. ...
... For solving SPPs, the most famous and commonly used layer-adapted meshes are Shishkin-type and Bakhvalov-type. Several methods have been studied for shiskin mesh, including FD methods [9], LDG method [10,11], and standard FEMs [12,13]; see their references for more details. Shishkin-type (S-type) meshes are widely used because of their straightforward structure and ability to provide precise analysis. ...
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.
... Singularly perturbed convection-diffusion problems constitute a class of mathematical models that arise in various fields of science and engineering [Schlichting and Gersten (2003); Naidu and Rao (1985); Miller (1997)]. These problems involve the simultaneous interaction of convection and diffusion phenomena, and a small parameter characterizes them, which multiplies the highest-order derivative term in the governing equations [Linß (2010)]. This small parameter introduces a significant disparity in the scales of the convection and diffusion processes, leading to boundary and interior layers. ...
... If the exact solution is unavailable, we obtain the maximum absolute error using the double mesh principle [Linß (2010)] given by ...
This paper presents a semi-analytical approach to solve a coupled system of singularly perturbed differential equations with shifts of mixed type. The solution to the problem manifests a distinctive multiscale nature, characterized by localized narrow regions where the solution undergoes exponential changes. Beyond these regions, the solutions exhibit smooth variations. We employ a factorization approach to address intricate multiscale characteristics, splitting the given coupled system into two explicit systems. The first, a degenerate system, captures the smooth solution outside boundary layers through initial value problems. The second, addressing solutions within boundary layers, utilizes stretching transformations to form a set of boundary value problems. Even though this factorization seems straightforward, the solutions obtained from these simplified systems capture the essential characteristics of the given system. The Runge–Kutta method is employed to solve the degenerate system of initial value problems, while the system of boundary value problems is solved analytically using asymptotic expansions. We establish the stability and consistency of the proposed method. The method converges uniformly with higher-order accuracy. The proposed method is easy to implement, does not require an adaptive mesh generation procedure to sustain approximation, and the estimates do not involve logarithmic terms. Numerical results and illustrations are presented for four model problems to underscore the effectiveness and potential of the presented approach.
... Shishkin meshes: These meshes are special piecewise uniform meshes that resolve the layer phenomena of the problem. To construct them, we define the transition parameters as follows (see [28]): ...
... Bakhvalov meshes: These meshes are graded in nature and superior to the Shishkin meshes. To discretize the problem, we construct the Bakhvalov meshes using the idea of the equidistribution principle for the monitor function (see [28]) where > 0, = 1, … , is a user-chosen constant. ...
This work considers the numerical approximation of linear and nonlinear singularly perturbed initial value coupled systems of first-order, for which the diffusion parameters at each equation of the system are distinct and also they can have a different order of magnitude. To do that, we use two efficient discretization methods, which combine the backward differences and an appropriate splitting by components. Both a priori and a posteriori error estimates are proved for the proposed discretization methods. The developed numerical methods are more computationally efficient than those classical methods used to solve the same type of coupled systems. Extensive numerical experiments strongly confirm in practice the theoretical results and corroborate the superior performance of the current approach compared with previous existing approaches.
... Keywords: time dependent singularly perturbed problem; weak Galerkin method; semi-discrete and fully discrete schemes; layer-adapted-meshes 5034 for some constants β 1 , β 2 , and c 0 . The parabolic convection-dominated problem (1.1) has been utilized in a broad range of applied mathematics and engineering including fluid dynamics, electrical engineering, and the transport problem [1,2]. ...
... Following [2], we list some famous adaptive meshes including S-type, Bakhvalov-Shishkin meshes (BS-mesh), and B-type in Table 1. ...
In this paper, we designed and analyzed a weak Galerkin finite element method on layer adapted meshes for solving the time-dependent convection-dominated problems. Error estimates for semi-discrete and fully-discrete schemes were presented, and the optimal order of uniform convergence has been obtained. A special interpolation was delicately designed based on the structures of the designed method and layer-adapted meshes. We provided various numerical examples to confirm the theoretical findings.
... The present analysis provides the proof of parameter uniform convergence of a hybrid scheme on Shishkin and Bakhvalov meshes within the same framework (similar to [32]). In this context, we want to note that the work in [30] for parameterized problems is limited to only Shishkin meshes. ...
... M B (η)dη ≤ C, we obtain from (2.20) that ϑ( N ) ≤ C N −1 for τ ≥ 2, cf.[32]. Consequently, Theorem 2.1 gives the following second-order uniform convergence resultmax ||u − U || N , λ − λ ≤ C N −2 .(3.7) ...
In this work, we present the a priori and a posteriori error analysis of a hybrid difference scheme for integral boundary value problems of nonlinear singularly perturbed parameterized form. The discretization for the nonlinear parameterized equation constitutes a hybrid difference scheme which is based on a suitable combination of the trapezoidal scheme and the backward difference scheme. Further, we employ the composite trapezoidal scheme for the discretization of the nonlocal boundary condition. A priori error estimation is provided for the proposed hybrid scheme, which leads to second-order uniform convergence on various a priori defined meshes. Moreover, a detailed a posteriori error analysis is carried out for the present hybrid scheme which provides a proper discretization of the error equidistribution at each partition. Numerical results strongly validate the theoretical findings for nonlinear problems with integral boundary conditions.
... The solution of the problem changes rapidly within layer region whereas it behaves slowly and regularly outside of the layer region. To overcome this complexity, robust discretizations are needed [12,21,23,26]. Some existence and uniqueness results about singularly perturbed problems have been given in [15,23]. ...
... [3,21,23,26,30,). For an even number N, we divide each of the subintervals [0, σ] and [σ, T ] into N 2 equidistant subintervals. ...
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.
... Table 13 provides a comparison between the results of the proposed method and SPPs. These methods often require intricate mesh refinement or specialized techniques to capture boundary layers, like implementation over layer adapted meshes [21](eg., Shishkin/Bakhvalov/Vulanovic meshes, etc) for capturing the boundary layers effectively, but that again depends on prior knowledge of the boundary layer's width and location, making the implementation cumbersome. Also, two-parameter SPPs exhibit varying solution behaviors depending on the ratio of the two parameters as discussed in Section 2 , further complicating the implementation of these numerical methods. ...
In this article, our goal is to solve two-parameter singular perturbation problems (SPPs) in one- and two-dimensions using an adapted Physics-Informed Neural Networks (PINNs) approach. Such problems are of major importance in engineering and sciences as it appears in control theory, fluid and gas dynamics, financial modelling and so on. Solutions of such problems exhibit boundary and/or interior layers, which make them difficult to handle. It has been validated in the literature that standard PINNs have low accuracy and can't handle such problems efficiently. Recently Cao et. al \cite{cao2023physics} proposed a new parameter asymptotic PINNs (PA-PINNs) to solve one-parameter singularly perturbed convection-dominated problems. It was observed that PA-PINNs works better than standard PINNs and gPINNs in terms of accuracy, convergence and stability. In this article, for the first time robustness of PA-PINNs will be validated for solving two-parameter SPPs.
... In particular, meshes for layers of exponential type have been examined, see e.g. [6] where various problems, numerical methods, and meshes are presented. Popular examples are, due to their simplicity, the piecewise equidistant Shishkin meshes [11,9] which are fine only in the layer region. ...
We consider singularly perturbed boundary value problems with a simple interior turning point whose solutions exhibit an interior layer. These problems are discretised using higher order finite elements on layer-adapted piecewise equidistant meshes proposed by Sun and Stynes. We also study the streamline-diffusion finite element method (SDFEM) for such problems. For these methods error estimates uniform with respect to are proven in the energy norm and in the stronger SDFEM-norm, respectively. Numerical experiments confirm the theoretical findings.
... The mesh that will be used here is a modified Shishkin mesh from [13,14], which has a greater smoothness compared to the generating function. Before the construction of the mesh, we are stating a theorem about the decomposition and estimates of the derivatives, which is necessary for the construction and further analysis. ...
We consider an approximate solution for the one-dimensional semilinear singularly-perturbed boundary value problem, using the previously obtained numerical values of the boundary value problem in the mesh points and the representation of the exact solution using Green's function. We present an -uniform convergence of such gained the approximate solutions, in the maximum norm of the order on the observed domain. After that, the constructed approximate solution is repaired and we obtain a solution, which also has --uniform convergence, but now of order on [0,1]. In the end a numerical experiment is presented to confirm previously shown theoretical results.
... Problems with a constant diffusion coefficient ε are widely studied in the literature: [2,4,12,15]. Problems with a variable diffusion coefficient (ε + x) p , p = 1 were formulated in the monograph by Polubarinova-Kochina [14] in order to model the filtration of liquid in the neighbourhood of a circular orifice with a small radius r = ε, while those with p = 2 appear in the physics of the motion of charges viewed as classical particles (see Zamaraev et al. [17]). ...
The paper discusses a two-point boundary value problem with a boundary turning point and a quadratic diffusion coefficient. It establishes bounds on solution derivatives, describes layer-eliminating coordinate transformations and corresponding layer-resolving grids, and analyses the convergence of numerical solutions by an upwind scheme on the layer-resolving grids.
... The case with a constant diffusion coefficient (d(x) = 0) in (1) is widely studied in the literature [8,16,20,21]. The problem of this nature, with d(x) = x was articulated in the monograph authored by Polubarinova-Kochina [18]. ...
... Numerical solutions to scalar singularly perturbed convection-diffusion problems have received considerable attention in research (e.g., [8,14,15,20,29]). Looking into finite element based numerical techniques, Kumar et al. [16] explored Crank-Nicolson weak Galerkin finite element method (FEM) for singularly perturbed unsteady convection-diffusion equation with a nonlinear reaction term in 2D. ...
... Further, a large number of mesh points are required to produce sufficiently precise approximations of the actual solution for various perturbation parameter values. Many methods have been developed to address this challenge, such as parameteruniform, parameter-robust, or uniformly convergent approaches like Fitted Operator Methods (FOMs) [7,8] and Fitted Mesh Methods (FMMs) [9,10]. Moreover, to address the challenges with meshes, various meshfree methods, such as the moving least squares (MLS) method [11], element-free Galerkin approach [12], and the local Petrov-Galerkin method [13], have been developed to solve SPPs numerically. ...
Physics-informed neural networks (PINNs) are a class of deep learning models that utilize physics as differential equations to address complex problems, including ones that may involve limited data availability. However, tackling solutions of differential equations with oscillations or singular perturbations and shock-like structures becomes challenging for PINNs. Considering these challenges, we designed an efficient wavelet-based PINNs (W-PINNs) model to solve singularly perturbed differential equations. Here, we represent the solution in wavelet space using a family of smooth-compactly supported wavelets. This framework represents the solution of a differential equation with significantly fewer degrees of freedom while still retaining in capturing, identifying, and analyzing the local structure of complex physical phenomena. The architecture allows the training process to search for a solution within wavelet space, making the process faster and more accurate. The proposed model does not rely on automatic differentiations for derivatives involved in differential equations and does not require any prior information regarding the behavior of the solution, such as the location of abrupt features. Thus, through a strategic fusion of wavelets with PINNs, W-PINNs excel at capturing localized nonlinear information, making them well-suited for problems showing abrupt behavior in certain regions, such as singularly perturbed problems. The efficiency and accuracy of the proposed neural network model are demonstrated in various test problems, i.e., highly singularly perturbed nonlinear differential equations, the FitzHugh-Nagumo (FHN), and Predator-prey interaction models. The proposed design model exhibits impressive comparisons with traditional PINNs and the recently developed wavelet-based PINNs, which use wavelets as an activation function for solving nonlinear differential equations.
... Для уравнений типа (1) первые формулы координатных преобразований с явной зависимостью от малого параметра предложены в работах [1][2][3]. На их основе были разработаны численные алгоритмы для решения задач с экспоненциальными слоями [4][5][6]. Однако указанные выше зависимости оказались пригодными исключительно для экспоненциальных слоев. Для неэкспоненциальных слоев (степенных, логарифмических, смешанных) первые результаты по построению координатных отображений с явной зависимостью от опубликованы в работах [7][8][9]. ...
Modified upwind and hybrid schemes on special grids for solving layered problems
V. I. Paasonen1,2,*
, V. D. Liseikin1
1Federal Research Center for Information and Computational Technologies, 630090, Novosibirsk, Russia
2Novosibirsk State University, 630090, Novosibirsk, Russia
*Corresponding author: Viktor I. Paasonen, e-mail: paas@ict.nsc.ru
Received March 03, 2023, revised August 10, 2023, accepted August 17, 2023.
Abstract
Boundary and interior layers present serious difficulties for the efficient calculation of equations
modelling many technical applications, in particular, those having a small parameter before the
higher derivatives. Due to this phenomenon, developing uniformly convergent algorithms for solving
such problems are difficult.
Resources provided by numerical schemes and adaptive grids can significantly reduce the adverse
effects on the accuracy of numerical experiments due to the layers. An efficient and popular scheme
for solving two-point singularly-perturbed problems with layers is the upwind difference scheme.
However, this scheme provides convergence of the first order only.
In this paper, we are focused on two second-order uniformly convergent finite difference algorithms
for solving two-point singularly-perturbed problems. The proposed algorithms apply a hybrid scheme
based on the midpoint upwind approximation, Buleev’s scheme and special layer-resolving grids
designed for solving problems with exponential and power layers of the first type.
Numerical experiments conducted out for singularly perturbed problems confirm the efficiency
of the algorithms for various values of the small parameter and show that the proposed method
provides competitive results compared to other methods available in the literature.
... Для уравнений типа (1) первые формулы координатных преобразований с явной зависимостью от малого параметра предложены в работах [1][2][3]. На их основе были разработаны численные алгоритмы для решения задач с экспоненциальными слоями [4][5][6]. Однако указанные выше зависимости оказались пригодными исключительно для экспоненциальных слоев. Для неэкспоненциальных слоев (степенных, логарифмических, смешанных) первые результаты по построению координатных отображений с явной зависимостью от опубликованы в работах [7][8][9]. ...
Recently, the Local Discontinuous Galerkin (LDG) method has been successfully applied to solve various reaction-diffusion problems characterized by boundary layers. To achieve uniform convergence in the small perturbation parameter , layer-adapted meshes of Shishkin-type along with their variants, have been frequently combined with the numerical schemes. However, limited convergence results have been reported for the LDG method applied to graded meshes. These meshes, generated through recursive formulae, present a compelling alternative, especially they are less sensitive to variations in , which determines the mesh structure and the solution behavior. In the present paper, we investigate the convergence of the LDG method on two Duran-type meshes: the Duran-Shishkin mesh and the Duran mesh. We derive optimal-order error estimates if the logarithmic factor is neglected. These results, appearing in the literature for the first time, open up new possibilities for the application of the LDG method with graded meshes in solving problems with boundary layers. Numerical results confirm that our theoretical estimates are sharp.
In this paper a numerical method on a posteriori mesh is presented to solve a second-order singularly perturbed convection-diffusion equation. A second-order three-point difference method is used to discretize the singularly perturbed convection-diffusion equation. A posteriori error analysis involving simple calculations with fewer proof techniques is developed for the second-order three-point difference method on an arbitrary mesh. A solution-adaptive algorithm based on a posteriori error analysis is designed to generate a posteriori mesh and the approximation solution. Numerical experiments verify the second-order uniform convergence of this method.
The local discontinuous Galerkin (LDG) method is used to solve a singularly perturbed convection–diffusion problem in one dimension. On two graded meshes, i.e., Duran–Shishkin and Duran meshes, we establish optimal-order error estimates of the LDG method in the energy norm, uniformly up to a logarithmic factor. Numerical experiments confirm the sharpness of our theoretical results.
The purpose of this chapter is twofold. First, it will summarize the most important open questions concerning the methods discussed in this monograph. And second, a few topics that are in some sense related to the contents of this monograph will be briefly addressed.
This chapter is the first one that studies discretizations of the convection-diffusion-reaction problem (2.8) that are monotone or obey DMPs under appropriate assumptions. In order to concentrate on the impact of convection, in particular if convection dominates, several simplifications will be taken into account, which will be described now.
The common way to improve properties of the Galerkin finite element discretization of a convection-diffusion-reaction problem is to modify the variational form, e.g., by changing the definition of some of the bilinear forms or by adding additional terms, as it was discussed in the previous chapters. However, it is also possible to modify directly the linear algebraic system corresponding to the Galerkin finite element discretization, which will be pursued in this chapter. To obtain accurate approximate solutions, these algebraic modifications have to be defined in a nonlinear way.
In this article, we apply the Direct Discontinuous Galerkin (DDG) method to compute the numerical solution of two-parameter singularly perturbed convection-diffusion-reaction boundary value problems. We establish the consistency, stability, and convergence of order k (up to a logarithmic factor) in an energy-type norm, where k represents the polynomial degree in the finite element space. Our theoretical findings are validated through numerical experiments. Keywords Two-parameter singularly perturbed boundary-value problems · Direct discontinuous Galerkin finite element method (DDGFEM) · Shishkin mesh · DG norm Mathematics Subject Classification 65L11 · 65L20 · 65L60 · 65L70
We consider a model convection-diffusion problem and present useful connections between the finite differences and finite element discretization methods. We introduce a general upwinding Petrov–Galerkin discretization based on bubble modification of the test space and connect the method with the general upwinding approach used in finite difference discretization. We write the finite difference and the finite element systems such that the two corresponding linear systems have the same stiffness matrices, and compare the right-hand side load vectors for the two methods. This new approach allows for improving well-known upwinding finite difference methods and for obtaining new error estimates. We prove that the exponential bubble Petrov–Galerkin discretization can recover the interpolant of the exact solution. As a consequence, we estimate the closeness of the related finite difference solutions to the interpolant. The ideas we present in this work, can lead to building efficient new discretization methods for multidimensional convection dominated problems.
For a model convection-diffusion problem, we obtain new error estimates for a general upwinding finite element discretization based on bubble modification of the test space. The key analysis tool is finding representations of the optimal norms on the trial spaces at the continuous and discrete levels. We analyze and compare three methods: the standard linear discretization, the saddle point least square and the upwinding Petrov–Galerkin methods. We conclude that the bubble upwinding the Petrov–Galerkin method is the most performant discretization for the one-dimensional model. Our results for the model convection-diffusion problem can be extended for creating new and efficient discretizations for the multi-dimensional cases.
In this article, we present a numerical method for the Stokes flow of an Oldroyd-B fluid. The viscoelastic stress evolves according to a constitutive law formulated in terms of the upper convected time derivative. A finite difference method is used to discretise along fluid trajectories to approximate the advection and deformation terms of the upper convected derivative in a simple, cheap and cohesive manner, as well as ensuring that the discrete conformation tensor is positive definite. A full implementation with coupling to the fluid flow is presented, along with a detailed discussion of the issues that arise with such schemes. We demonstrate the performance of this method with detailed numerical experiments in a lid-driven cavity setup. Numerical results are benchmarked against published data, and the method is shown to perform well in this challenging case.
A singularly perturbed reaction-diffusion problem posed on the unit square in R 2 \mathbb {R}^2 is solved numerically by a local discontinuous Galerkin (LDG) finite element method. Typical solutions of this class of 2D problems exhibit boundary layers along the sides of the domain; these layers generally cause difficulties for numerical methods. Our LDG method handles the boundary layers by using a Shishkin mesh and also introducing the new concept of a “layer-upwind flux”—a discrete flux whose values are chosen on the fine mesh (which lies inside the boundary layers) in the direction where the layer weakens. On the coarse mesh, one can use a standard central flux. No penalty terms are needed with these fluxes, unlike many other variants of the LDG method. Our choice of discrete flux makes it feasible to derive an optimal-order error analysis in a balanced norm; this norm is stronger than the usual energy norm and is a more appropriate measure for errors in computed solutions for singularly perturbed reaction-diffusion problems. It will be proved that the LDG method is usually convergent of order O ( ( N − 1 ln N ) k + 1 ) O((N^{-1}\ln N)^{k+1}) in the balanced norm, where N N is the number of mesh intervals in each coordinate direction and tensor-product piecewise polynomials of degree k k in each coordinate variable are used in the LDG method. This result is the first of its kind for the LDG method applied to this class of problem and is optimal for convergence on a Shishkin mesh. Its sharpness is confirmed by numerical experiments.
So far uniform pointwise error estimate of the local discontinuous Galerkin (LDG) method for the singularly perturbed reaction-diffusion problems is only known in the one-dimensional case. In this paper, we consider an LDG method with interior alternating numerical flux for a singularly perturbed reaction-diffusion problem posed on the unit square in . We prove that the LDG method on a Shishkin mesh is pointwise convergent with the order of , uniformly in the perturbation parameter outside the layers, where k is the degree of the tensor-product piecewise polynomials used in the finite element space and N is the number of mesh elements in each coordinate direction. This rate of convergence is sharp and agrees with our numerical results. In the fine parts that lie inside the boundary and corner layers, we derive a uniform pointwise convergence of orders and respectively. The theoretical difficulty lies in overcoming the highly non-uniformity of the mesh and the weak stability induced by the bilinear form. Numerical experiments are also given.
This article presents a numerical solution for a specific class of 2D parabolic singularly perturbed convection-diffusion problems with a special interior line source. The proposed approach employs the alternating direction implicit type operator splitting streamline-diffusion finite element method (SDFEM), offering a viable solution to alleviate computational complexity and high storage requirements encountered in higher-dimensional problems. The overall stability of the two-step method is established, while a piecewise-uniform Shishkin mesh is employed for spatial domain discretization. By carefully selecting the stabilization parameter, an -uniform error estimate is derived, accounting for the influence of the time step-interval which is essential to maintain the method’s stability. To validate the theoretical error estimate, numerical investigation are conducted, showcasing the effectiveness of the proposed method. This research contributes to advancing the understanding and numerical treatment of this specific class of 2D parabolic singularly perturbed convection-diffusion problems, shedding light on the intricate dynamics and behavior of the system in the presence of a special interior line source.
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 study, we investigate a hybrid-type anisotropic weakly over-penalised symmetric interior penalty method for the Poisson equation on convex domains. Compared with the well-known hybrid discontinuous Galerkin methods, our approach is simple and easy to implement. Our primary contributions are the proposal of a new scheme and the demonstration of a proof for the consistency term, which allows us to estimate the anisotropic consistency error. The key idea of the proof is to apply the relation between the Raviart–Thomas finite element space and a discontinuous space. In numerical experiments, we compare the calculation results for standard and anisotropic mesh partitions.
A singularly perturbed reaction–diffusion problem in 1D is solved numerically by a local discontinuous Galerkin (LDG) finite element method. For this type of problem the standard energy norm is too weak to capture the contribution of the boundary layer component of the true solution, so balanced norms have been used by many authors to give more satisfactory error bounds for solutions computed using various types of finite element method. But for the LDG method, up to now no optimal-order balanced-norm error estimate has been derived. In this paper, we consider an LDG method with central numerical flux on a Shishkin mesh. Using the superconvergence property of the local L2 projector and some local coupled projections around the two transition points of the mesh, we prove an optimal-order balanced-norm error estimate for the computed solution; that is, when piecewise polynomials of degree k are used on a Shishkin mesh with N mesh intervals, in the balanced norm we establish O((N-1lnN)k+1) convergence when k is even and O((N-1lnN)k) when k is odd. Numerical experiments confirm the sharpness of these error bounds.
In this study, we investigate an anisotropic weakly over-penalised symmetric interior penalty method for the Stokes equation on convex domains. Our approach is a simple discontinuous Galerkin method similar to the Crouzeix–Raviart finite element method. As our primary contribution, we show a new proof for the consistency term, which allows us to obtain an estimate of the anisotropic consistency error. The key idea of the proof is to apply the relation between the Raviart–Thomas finite element space and a discontinuous space. While inf-sup stable schemes of the discontinuous Galerkin method on shape-regular mesh partitions have been widely discussed, our results show that the Stokes element satisfies the inf-sup condition on anisotropic meshes. Furthermore, we provide an error estimate in an energy norm on anisotropic meshes. In numerical experiments, we compare calculation results for standard and anisotropic mesh partitions.
We present a parameter-robust finite difference method for solving a system of weakly coupled singularly perturbed convection-diffusion equations. The diffusion coefficient of each equation is a small distinct positive parameter. Due to this, the solution to the system has, in general, overlapping boundary layers. The problem is discretized using a particular combination of a compact second-order difference scheme and a central difference scheme on a piecewise-uniform Shishkin mesh. The convergence analysis is given, and the method is shown to have almost second-order uniform convergence in the maximum norm with respect to the perturbation parameters. The results of numerical experiments are in agreement with the theoretical outcomes.
ResearchGate has not been able to resolve any references for this publication.