Natalia Kopteva

University of Limerick | UL · Department of Mathematics and Statistics (MACSI)

We prove residual-type a posteriori error estimates in the maximum norm for a linear scalar elliptic convection–diffusion problem that may be singularly perturbed. Similar error analysis in the energy norm by Verfürth indicates that a dual norm of the convective derivative of the error must be added to the natural energy norm in order for the natur...

A posteriori error estimates in the maximum norm are studied for various time-semidiscretisations applied to a class of linear parabolic equations. We summarise results from the literature and present some new improved error bounds. Crucial ingredients are certain bounds in the $L_1$ norm for the Green's function associated with the parabolic opera...

This paper presents an extended version of the article [Franz, S., Kopteva, N.: J. Differential Equations, 252 (2012)]. The main improvement compared to the latter is in that here we additionally estimate the mixed second-order derivative of the Green's function. The case of Neumann conditions along the characteristic boundaries is also addressed....

Time-fractional parabolic equations with a Caputo time derivative are considered. For such equations, we explore and further develop the new methodology of the a-posteriori error estimation and adaptive time stepping proposed in [7]. We improve the earlier time stepping algorithm based on this theory, and specifically address its stable and efficie...

An initial-boundary value problem of subdiffusion type is considered; the temporal component of the differential operator has the form \(\sum _{i=1}^{\ell }q_i(t)\, D _t ^{\alpha _i} u(x,t)\), where the \(q_i\) are continuous functions, each \(D _t ^{\alpha _i}\) is a Caputo derivative, and the \(\alpha _i\) lie in (0, 1]. Maximum/comparison princi...

We prove residual-type a posteriori error estimates in the maximum norm for a linear scalar elliptic convection-diffusion problem that may be singularly perturbed. Similar error analysis in the energy norm by Verf\"{u}rth indicates that a dual norm of the {convective derivative of the} error must be added to the natural energy norm in order for the...

We consider time-fractional parabolic equations with a Caputo time derivative of order α∈(0,1). For such equations, we give an elementary proof of the weak maximum principle under no assumptions on the sign of the reaction coefficient. This proof is also extended for weak solutions, as well as for various types of boundary conditions and variable-c...

An initial-boundary value problem of subdiffusion type is considered; the temporal component of the differential operator has the form $\sum_{i=1}^{\ell}q_i(t)\, D _t ^{\alpha_i} u(x,t)$, where the $q_i$ are continuous functions, each $D _t ^{\alpha_i}$ is a Caputo derivative, and the $\alpha_i$ lie in $(0,1]$. Maximum/comparison principles for thi...

We consider time-fractional parabolic equations with a Caputo time derivative of order $\alpha\in(0,1)$. For such equations, we give an elementary proof of the weak maximum principle under no assumptions on the sign of the reaction coefficient. This proof is also extended for weak solutions, as well as for various types of boundary conditions and v...

For time-fractional parabolic equations with a Caputo time derivative of order α∈(0,1), we give pointwise-in-time a posteriori error bounds in the spatial L2 and L∞ norms. Hence, an adaptive mesh construction algorithm is applied for the L1 method, which yields optimal convergence rates 2−α in the presence of solution singularities.

For time-fractional parabolic equations with a Caputo time derivative of order $\alpha\in(0,1)$, we give pointwise-in-time a posteriori error bounds in the spatial $L_2$ and $L_\infty$ norms. Hence, an adaptive mesh construction algorithm is applied for the L1 method, which yields optimal convergence rates $2-\alpha$ in the presence of solution sin...

Lower a posteriori error bounds obtained using the standard bubble function approach are reviewed in the context of anisotropic meshes. A numerical example is given that clearly demonstrates that the short-edge jump residual terms in such bounds are not sharp. Hence, for linear finite element approximations of the Laplace equation in polygonal doma...

An initial-boundary value problem with a Caputo time derivative of fractional order α ∈ ( 0 , 1 ) \alpha \in (0,1) is considered, solutions of which typically exhibit a singular behaviour at an initial time. An L2-type discrete fractional-derivative operator of order 3 − α 3-\alpha is considered on nonuniform temporal meshes. Sufficient conditions...

A semilinear initial-boundary value problem with a Caputo time derivative of fractional order $\alpha\in(0,1)$ is considered, solutions of which typically exhibit a singular behaviour at an initial time. For L1-type discretizations of this problem, we employ the method of upper and lower solutions to obtain sharp pointwise-in-time error bounds on q...

In Kopteva (2014) a counterexample of an anisotropic triangulation was given on which the exact solution has a second-order error of linear interpolation, while the computed solution obtained using linear finite elements is only first-order pointwise accurate. This example was given in the context of a singularly perturbed reaction–diffusion equati...

Lower a posteriori error bounds obtained using the standard bubble function approach are reviewed in the context of anisotropic meshes. A numerical example is given that clearly demonstrates that the short-edge jump residual terms in such bounds are not sharp. Hence, for linear finite element approximations of the Laplace equation in polygonal doma...

Lower a posteriori error bounds obtained using the standard bubble function approach are reviewed in the context of anisotropic meshes. A numerical example is given that clearly demonstrates that the short-edge jump residual terms in such bounds are not sharp. Hence, for linear finite element approximations of the Laplace equation in polygonal doma...

An initial-boundary value problem with a Caputo time derivative of fractional order α∈(0,1) is considered, solutions of which typically exhibit a singular behaviour at an initial time. For this problem, we give a simple and general numerical-stability analysis using barrier functions, which yields sharp pointwise-in-time error bounds on quasi-grade...

An initial-boundary value problem with a Caputo time derivative of fractional order α∈(0,1) is considered, solutions of which typically exhibit a singular behaviour at an initial time. For this problem, we give a simple and general numerical-stability analysis using barrier functions, which yields sharp pointwise-in-time error bounds on quasi-grade...

An initial-boundary value problem with a Caputo time derivative of fractional order $\alpha\in(0,1)$ is considered, solutions of which typically exhibit a singular behaviour at an initial time. An L2-type discrete fractional-derivative operator of order $3-\alpha$ is considered on nonuniform temporal meshes. Sufficient conditions for the inverse-mo...

In [Kopteva, Math. Comp., 2014] a counterexample of an anisotropic triangulation was given on which the exact solution has a second-order error of linear interpolation, while the computed solution obtained using linear finite elements is only first-order pointwise accurate. This example was given in the context of a singularly perturbed reaction-di...

In the recent article [Kopteva, N., Numer. Math., 137, 607--642 (2017)] the author obtained residual-type a posteriori error estimates in the energy norm for singularly perturbed semilinear reaction-diffusion equations on anisotropic triangulations. The error constants in these estimates are independent of the diameters and the aspect ratios of mes...

Residual-type a posteriori error estimates in the energy norm are given for singularly perturbed semilinear reaction-diffusion equations posed in polygonal domains. Linear finite elements are considered on anisotropic triangulations. The error constants are independent of the diameters and the aspect ratios of mesh elements and of the small perturb...

For linear finite element discretizations of the Laplace equation in three dimensions, we give an example of a tetrahedral mesh in the cubic domain for which the logarithmic factor cannot be removed from the standard upper bounds on the error in the maximum norm.

Linear and semilinear second-order parabolic equations are considered. For these equations, we give a posteriori error estimates in the maximum norm that improve upon recent results in the literature. In particular it is shown that logarithmic dependence on the time step size can be eliminated. Semidiscrete and fully discrete versions of the backwa...

An initial-boundary value problem with a Caputo time derivative of fractional order $\alpha\in(0,1)$ is considered, solutions of which typically exhibit a singular behaviour at an initial time. For this problem, we give a simple framework for the analysis of the error of L1-type discretizations on graded and uniform temporal meshes in the $L_\infty...

Fully computable a posteriori error estimates in the energy norm are given for singularly perturbed semilinear reaction-diffusion equations posed in polygonal domains. Linear finite elements are considered on anisotropic triangulations. To deal with the latter, we employ anisotropic quadrature and explicit anisotropic flux reconstruction. Prior to...

Residual-type a posteriori error estimates in the energy norm are given for singularly perturbed semilinear reaction-diffusion equations posed in polygonal domains. Linear finite elements are considered on anisotropic triangulations. The error constants are independent of the diameters and the aspect ratios of mesh elements and of the small perturb...

A two-point boundary value problem is considered on the interval [0, 1], where the leading term in the differential operator is a Riemann-Liouville fractional derivative of order 2 − δ with 0 < δ < 1. It is shown that any solution of such a problem can be expressed in terms of solutions to two associated weakly singular Volterra integral equations...

Residual-type a posteriori error estimates in the maximum norm are given for singularly perturbed semilinear reaction-diffusion equations posed in polyhedral domains. Standard finite element approximations are considered. The error constants are independent of the diameters of mesh elements and the small perturbation parameter. In our analysis, we...

Residual-type a posteriori error estimates in the maximum norm are given for singularly perturbed semilinear reaction-diffusion equations posed in polygonal domains. Linear finite elements are considered on anisotropic triangulations. The error constants are independent of the diameters and the aspect ratios of mesh elements and of the small pertur...

A two-point boundary value problem is considered on the interval $[0,1]$, where the leading term in the differential operator is a Riemann-Liouville fractional derivative of order $2-\delta$ with $0<\delta<1$. It is shown that any solution of such a problem can be expressed in terms of solutions to two associated weakly singular Volterra integral e...

We give a counterexample of an anisotropic triangulation on which the exact solution has a second-order error of linear interpolation, while the computed solution obtained using linear finite elements is only first-order pointwise accurate. Our example is given in the context of a singularly perturbed reaction-diffusion equation, whose exact soluti...

[A revised version of this paper will appear in BIT: DOI 10.1007/s10543-014-0539-4] A two-point boundary value problem is considered on the interval [0, 1], where the leading term in the differential operator is a Caputo fractional-order derivative of order 2 − δ with 0 < δ < 1. The problem is reformulated as a Volterra integral equation of the sec...

A semilinear second-order parabolic equation is considered in a regular and a singularly perturbed regime. For this equation, we give computable a posteriori error estimates in the maximum norm. Semidiscrete and fully discrete versions of the backward Euler, Crank-Nicolson, and discontinuous Galerkin dG(r) methods are addressed. For their full disc...

A second-order singularly perturbed parabolic equation in one space dimension is considered. For this equation, we give computable a posteriori error estimates in the maximum norm for two semidiscretisations in time and a full discretisation using P 1 FEM in space. Both the Backward-Euler method and the Crank-Nicolson method are considered. Certain...

A semilinear second-order singularly perturbed parabolic equation in one space dimension is considered. For this equation, we give computable a posteriori error estimates in the maximum norm for a difference scheme that uses Backward-Euler in time and central differencing in space. Sharp L¹-norm bounds for the Green's function of the parabolic oper...

We consider a singularly perturbed convection–diffusion problem posed in the unit square with a horizontal convective direction. Its solutions exhibit parabolic and exponential boundary layers. Sharp estimates of the Greenʼs function and its first- and second-order derivatives are derived in the L1L1 norm. The dependence of these estimates on the s...

An overlapping Schwarz domain decomposition is applied to a semilinear reaction-diffusion equation posed in a smooth two-dimensional domain. The problem may exhibit multiple solutions; its diffusion parameter ε 2 is arbitrarily small, which induces boundary layers. The Schwarz method invokes a boundary-layer subdomain and an interior subdomain, the...

A semilinear reaction–diffusion two-point boundary value problem, whose second-order derivative is multiplied by a small positive parameter e2{\varepsilon^2} , is considered. It can have multiple solutions. The numerical computation of solutions having interior transition layers is analysed. It is demonstrated that the accurate computation of such...

A singularly perturbed semilinear reaction-diffusion problem in the unit cube, is discretized on arbitrary nonuniform tensor-product meshes. We establish a second-order maximum norm a posteriori error estimate that holds true uniformly in the small diffusion parameter. No mesh aspect ratio condition is imposed. This result is obtained by combining...

An initial boundary-value problem for a semilinear reaction–diffusion equation is considered. Its diffusion parameter ϵ2 is arbitrarily small, which induces initial and boundary layers. It is shown that the conventional implicit method might produce incorrect computed solutions on uniform meshes. Therefore we propose a stabilized method that yields...

A linear singularly perturbed convection-diffusion problem with characteristic layers is considered in three dimensions. Sharp bounds for the associated Green's function and its derivatives are established in the $L_1$ norm. The dependence of these bounds on the small perturbation parameter is shown explicitly. The obtained estimates will be used i...

Linear singularly perturbed convection-diffusion problems with characteristic layers are considered in three dimensions. We demonstrate the sharpness of our recently obtained upper bounds for the associated Green's function and its derivatives in the $L_1$ norm. For this, in this paper we establish the corresponding lower bounds. Both upper and low...

A linear singularly perturbed convection-diffusion problem with characteristic layers is considered in three dimensions. Sharp bounds for the associated Green's function and its first-and second-order derivatives are established in the L 1 norm by employing the parametrix method. The dependence of these bounds on the small perturbation parameter is...

A semilinear reaction-diffusion two-point boundary value problem, whose second-order derivative is multiplied by a small positive parameter $\eps^2$, is considered. It can have multiple solutions. An asymptotic expansion is constructed for a solution that has an interior layer. Further properties are then established for a perturbation of this expa...

The semilinear reaction–diffusion equation −ε2Δu+b(x,u)=0 with Dirichlet boundary conditions is considered in a convex polygonal domain. The singular perturbation parameter ε is arbitrarily small, and the “reduced equation”b(x,u0(x))=0 may have multiple solutions. An asymptotic expansion for u is constructed that involves boundary and corner layer...

We consider a two-point boundary-value problem for a singularly perturbed convection-diffusion problem. The problem is solved by using a defect-correction method based on a first-order upwind difference scheme and a second-order (unstabilized) central difference scheme. A robust a posteriori error estimate in the maximum norm is derived. It provide...

This article reviews some of the salient features of the piecewise-uniform Shishkin mesh. The central analytical techniques involved in the associated numerical analysis are explained via a particular class of singularly perturbed differential equations. A detailed discussion of the Shishkin solution decomposition is included. The generality of the...

A semilinear singularly perturbed reaction-diffusion equation with Dirichlet boundary conditions is considered in a convex unbounded sector. The singular perturbation parameter is arbitrarily small, and the "reduced equation" may have multiple solutions. A formal asymptotic expansion for a possible solution is constructed that involves boundary and...

The numerical solution of a singularly perturbed semilinear reaction–diffusion two-point boundary-value problem is addressed. The method considered is adaptive movement of a fixed number (N + 1) of mesh points by equidistribution of a monitor function that uses discrete second-order derivatives. We extend the analysis by Kopteva & Stynes (2001, SIA...

An overlapping Schwarz domain decomposition is applied to a semilinear reaction-diffusion two-point boundary value problem with multiple solutions. Its diffusion parameter ε2 is arbitrarily small, which induces boundary layers. The Schwarz method invokes two boundary-layer subdomains and an interior subdomain, the narrow overlapping regions being o...

In convection-diffusion problems, transport processes dominate while diffusion effects are confined to a relatively small part of the domain. This state of affairs means that one cannot rely on the formal ellipticity of the differential operator to ensure the convergence of standard numerical algorithms. Thus new ideas and approaches are required....

A singularly perturbed reaction-diffusion equation is posed in a two-dimensional L -shaped domain Omega subject to a continuous Dirchlet boundary condition. Its solutions are in the Hoelder space C^{2/3}(barOmega) and typically exhibit boundary layers and corner singularities. The problem is discretized on a tensor-product Shishkin mesh that is fur...

A semilinear reaction-diffusion equation with multiple solutions is considered in a smooth two-dimensional domain. Its diffusion parameter ε 2 is arbitrarily small, which induces boundary layers. We extend the numerical method and its maximum norm error analysis of the paper [N. Kopteva: Math. Comp. 76 (2007) 631–646], in which a parametrization of...

We examine steady flows of a thin film of viscous fluid on the inside of a cylinder with horizontal axis, rotating about this axis. If the amount of fluid in the cylinder is sufficiently small, all of it is entrained by rotation and the film is distributed more or less evenly. For medium amounts, the fluid accumulates on the ‘rising’ side of the cy...

A singularly perturbed semilinear reaction-diffusion equation, posed in the unit square, is discretized on arbitrary nonuniform tensor-product meshes. We establish a second-order maximum norm a posteriori error estimate that holds true uniformly in the small diffusion parameter. No mesh aspect ratio assumption is made. Numerical results are present...

A semilinear reaction-diffusion equation with multiple solutions is considered in a smooth two-dimensional domain. Its diffusion parameter \varepsilon^2 is arbitrarily small, which induces boundary layers. Constructing discrete sub- and super-solutions, we prove existence and investigate the accuracy of multiple discrete solutions on layer-adapted...

We consider a deterministic model of landscape evolution through the mechanism of overland flow over an erodible substrate, using the St. Venant equations of hydraulics together with the Exner equation for hillslope erosion. A novelty in the model is the allowance for a nonzero bedload layer thickness, which is necessary to distinguish between tran...

A singularly perturbed semilinear two-point boundary-value problem is discretized on arbitrary non-uniform meshes. We present second-order maximum norm a posteriori error estimates that hold true uniformly in the small parameter. Their application to monitor-function equidistribution and a posteriori mesh refinement are discussed. Numerical results...

Semilinear reaction-diffusion two-point boundary value problems with multiple solutions are considered. Here the second-order derivative is multiplied by a small positive parameter and consequently these solutions can have boundary or interior layers. A survey is given of the results obtained in our recent investigations into the numerical solution...

A semilinear reaction-diffusion equation with multiple solu-tions is considered in a smooth two-dimensional domain. Its diffusion parameter ε 2 is arbitrarily small, which induces boundary layers. We extend the numerical method and its maximum norm error analysis of the paper [N. Kopteva: Maximum norm error analysis of a 2d singularly perturbed sem...

We examine the stability of a thin film of viscous fluid inside a cylinder with horizontal axis, rotating about this axis. Depending on the parameters involved, the dynamics of the film can be described by several asymptotic equations, one of which was examined by Benilov, O'Brien, and Sazonov (J. Fluid Mech. 2003 497, 201–224). It turned out that...

The numerical solution of a linear singularly-perturbed reaction–diffusion two-point boundary value problem is considered. The method used is adaptive movement of a fixed number of mesh points by monitor-function equidistribution. A partly heuristic argument based on truncation error analysis leads to several suitable monitor functions, but also sh...

Two model two-dimensional singularly perturbed convection–diffusion problems are considered whose solutions may have characteristic boundary and interior layers. They are solved numerically by the streamline-diffusion finite element method using piecewise linear or bilinear elements. We investigate how accurate the computed solution is in character...

A nonlinear reaction–diffusion two-point boundary value problem with multiple solutions is considered. Its second-order derivative is multiplied by a small positive parameter ɛ, which induces boundary layers. Using dynamical systems techniques, asymptotic properties of its discrete sub- and super-solutions are derived. These properties are used to...

We consider a singularly perturbed convection-diffusion problem in a rectangular domain. It is solved numeric ally using a first-order upwind finite-difference scheme on a tensor-product piecewise-uniform Shishkin mesh with O(N) mesh points in each coordinate direction. It is known [G. I. Shishkin, Grid Approximations of Singularly Perturbed Ellipt...

A singularly perturbed quasilinear two-point boundary value problem with an exponential boundary layer is considered. The problem is discretized using the standard central difference scheme on generalized Shishkin-type meshes. We give a uniform second-order error estimate in a discrete L∞ norm. Numerical experiments support the theoretical results.

We consider a convection–diffusion two-point boundary value problem in conservative form. To solve it numerically an upwind conservative finite difference scheme is applied. On an arbitrary mesh we prove bounds, which are weighted by the small diffusion coefficient, on the errors in approximating the derivative of the true solution by divided diffe...

A singularly perturbed quasi-linear two-point boundary value problem with an exponential boundary layer is discretized on arbitrary nonuniform meshes using first- and second-order difference schemes, including upwind schemes. We give first- and second-order maximum norm a posteriori error estimates that are based on difference derivatives of the nu...

We consider two convection-diffusion boundary value problems in conservative form: for an ordinary differential equation and for a parabolic equation. Both the problems are discretized using a four-point second-order upwind space difference operator on arbitrary and layer-adapted space meshes. We give ɛ-uniform maximum norm error estimates O(N −2...

A quasi-linear conservative convection-diffusion two-point boundary value problem is considered. To solve it numerically, an upwind finite difference scheme is applied. The mesh used has a fixed number (N + 1) of nodes and is initially uniform, but its nodes are moved adaptively using a simple algorithm of de Boor based on equidistribution of the a...

For a singularly perturbed one-dimensional time-independent divergence equation of diffusion-convection, a scheme is analyzed that approximates the first-order derivative by the central difference. It is proved that this scheme is uniformly convergent with respect to a small parameter in the difference norm L∞h on the Bakhvalov and Shishkin grids r...

*** This file is available at http://www.staff.ul.ie/natalia/pubs.html *** For a singularly perturbed one-dimensional stationary divergent convection-diffusion equation, we consider the set of inhomogeneous monotone three-point finite-difference schemes on Bakhvalov and Shishkin grids that are condensed in the boundary layer and prove that the con...

*** This file is available at http://www.staff.ul.ie/natalia/pubs.html ***

***The file is available at http://www.staff.ul.ie/natalia/pubs.html*** A difference scheme with approximation of the first derivative by the central difference ratio is investigated for a singularly perturbed elliptic equation for the second order in the strip [0, 1]×(-∞, ∞) with periodicity conditions in the y variable. It is shown that for the...

***The file is available at http://www.staff.ul.ie/natalia/pubs.html***

***The file is available at http://www.staff.ul.ie/natalia/pubs.html*** A weighted two-layer difference scheme for the one-dimensional time-dependent convec-tion-diffusion equation is examined. In this scheme, the first spatial derivative is approximated by the central divided difference, It is shown that, on the Shishkin piecewise uniform grid con...

***The file is available at http://www.staff.ul.ie/natalia/pubs.html***

***The file is available at http://www.staff.ul.ie/natalia/pubs.html***

***The file is available at http://www.staff.ul.ie/natalia/pubs.html*** For an ordinary second-order differential equation in which the coefficient of the highest derivative is a small parameter, the classical difference scheme which uses a central difference ratio to approximate the first derivative is investigated. By means of a detailed analysi...