Layer-adapted Meshes Research Articles

Local projection methods on layer-adapted meshes for higher order discretisations of convection–diffusion problems

We consider singularly perturbed convection–diffusion problems in the unit square where the solutions show the typical exponential layers. Layer-adapted meshes (Shishkin and Bakhvalov–Shishkin meshes) and the local projection method are used to stabilise the discretised problem. Using enriched Q r -elements on the coarse part of the mesh and standard Q r -elements on the remaining parts of the mesh, we show that the difference between the solution of the stabilised discrete problem and a special interpolant of the solution of the continuous problem convergences ε-uniformly with order O ( N − ( r + 1 / 2 ) ) on Bakhvalov–Shishkin meshes and with order O ( N − ( r + 1 / 2 ) + N − ( r + 1 ) ln r + 3 / 2 N ) on Shishkin meshes. Furthermore, an ε-uniform convergence in the ε-weighted H 1 -norm with order O ( ( N − 1 ln N ) − r ) on Shishkin meshes and with order O ( N − r ) on Bakhvalov–Shishkin meshes will be proved. Numerical results which support the theory will be presented.

Continuous–discontinuous finite element method for convection-diffusion problems with characteristic layers

We study convergence properties of a numerical method for convection-diffusion problems with characteristic layers on a layer-adapted mesh. The method couples standard Galerkin with an h -version of the nonsymmetric discontinuous Galerkin finite element method with bilinear elements. In an associated norm, we derive the error estimate as well as the supercloseness result that are uniform in the perturbation parameter. Applying a post-processing operator for the discontinuous Galerkin method, we construct a new numerical solution with enhanced convergence properties.

Analysis of a FEM for a coupled system of singularly perturbed reaction–diffusion equations

We consider a system of l ≥ 2 one-dimensional singularly perturbed reaction–diffusion equations coupled at the zero-order term. The second derivative of each equation is multiplied by a distinct small parameter. We present a convergence theory for conforming linear finite elements on arbitrary meshes. As a result convergence independently of the perturbation parameters on a wide class of layer-adapted meshes is established.

A robust discontinuous Galerkin method for solving convection-diffusion problems

In this paper, a new DG method was designed to solve the model problem of the one-dimensional singularly-perturbed convection-diffusion equation. With some special chosen numerical traces, the existence and uniqueness of the DG solution is provided. The superconvergent points inside each element are observed. Particularly, the 2p + 1-order superconvergence and even uniform superconvergence under layer-adapted mesh are observed numerically.

A finite difference method on layer-adapted meshes for an elliptic reaction-diffusion system in two dimensions

An elliptic system of M(> 2) singularly perturbed linear reaction-diffusion equations, coupled through their zero-order terms, is considered on the unit square. This system does not in general satisfy a maximum principle. It is solved numerically using a standard difference scheme on tensor-product Bakhvalov and Shishkin meshes. An error analysis for these numerical methods shows that one obtains nodal O(N -2 ) convergence on the Bakhvalov mesh and O(N -2 ln 2 N) convergence on the Shishkin mesh, where N mesh intervals are used in each coordinate direction and the convergence is uniform in the singular perturbation parameter. The analysis is much simpler than previous analyses of similar problems, even in the case of a single reaction-diffusion equation, as it does not require the construction of an elaborate decomposition of the solution. Numerical results are presented to confirm our theoretical error estimates.

Layer-adapted meshes for a linear system of coupled singularly perturbed reaction-diffusion problems

We consider a system of l ≥ 2 one-dimensional singularly perturbed reaction-diffusion equations coupled at the zero-order term. The second derivative of each equation is multiplied by a distinct small parameter. We show how to decompose the solution to the problem into regular and layer parts. Properties of the discretized operator are established using discrete Green's functions. We prove that a central difference scheme on certain layer-adapted meshes converges independently of the perturbation parameters. Supporting numerical examples confirm our theoretical results.

Maximum-norm error analysis of a non-monotone FEM for a singularly perturbed reaction-diffusion problem

A non-monotone FEM discretization of a singularly perturbed one-dimensional reaction-diffusion problem whose solution exhibits strong layers is considered. The method is shown to be maximum-norm stable although it is not inverse monotone. Both a priori and a posteriori error bounds in the maximum norm are derived. The a priori result can be used to deduce uniform convergence of various layer-adapted meshes proposed in the literature. Numerical experiments complement the theoretical results.

Layer-adapted meshes and FEM for time-dependent singularly perturbed reaction-diffusion problems

We consider a non-monotone FEM discretisation of a singularly perturbed time-dependent reaction-diffusion problem whose solution exhibits strong parabolic layers. We conduct a stability and convergence analysis for arbitrary meshes. The method is shown to be maximum-norm stable although it is not inverse monotone. The convergence result implies convergence of the method on Shishkin and Bakhvalov meshes uniformly in the perturbation parameter e. Numerical experiments complement our theoretical results.

Maximum norm error analysis of a 2d singularly perturbed semilinear reaction-diffusion problem

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 meshes of Bakhvalov and Shishkin types. It is shown that one gets second-order convergence (with, in the case of the Shishkin mesh, a logarithmic factor) in the discrete maximum norm, uniformly in $\varepsilon$ for $\varepsilon \le Ch$. Here $h>0$ is the maximum side length of mesh elements, while the number of mesh nodes does not exceed $Ch^{-2}$. Numerical experiments are performed to support the theoretical results.

Analysis of an upwind finite-difference scheme for a system of coupled singularly perturbed convection-diffusion equations

We study a system of coupled convection-diffusion equations. The equations have diffusion parameters of different magnitudes associated with them which give rise to boundary layers at either boundary. An upwind finite difference scheme on arbitrary meshes is used to solve the system numerically. A general error estimate is derived that allows to immediately conclude robust convergence – w.r.t. the perturbation parameters – for certain layer-adapted meshes, thus improving and generalising previous results [4]. We present the results of numerical experiments to illustrate our theoretical findings.

Layer‐adapted meshes for time‐dependent reaction‐diffusion problems

Layer‐adapted meshes for time‐dependent reaction‐diffusion problems

Numerical analysis of singularly perturbed nonlinear reaction-diffusion problems with multiple solutions

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 of these problems on layer-adapted meshes.

Sufficient conditions for uniform convergence on layer-adapted meshes for one-dimensional reaction–diffusion problems

We study convergence properties of a finite element method with lumping for the solution of linear one-dimensional reaction–diffusion problems on arbitrary meshes. We derive conditions that are sufficient for convergence in the L∞ norm, uniformly in the diffusion parameter, of the method. These conditions are easy to check and enable one to immediately deduce the rate of convergence. The key ingredients of our analysis are sharp estimates for the discrete Green function associated with the discretization.

Exponentially-fitted methods on layer-adapted meshes

In this paper, a new derivation of a uniformly-convergent, second-order method for singularly-perturbed, linear ordinary differential equations based on the freezing of the coefficients of the differential equation, and integration of the resulting equations subject to continuity and smoothness conditions at the nodes, is presented. The derivation presented here is compared with others based on Green’s functions, when only advection and diffusion processes are considered when solving the homogeneous equations. In addition, a new method that accounts for advection, diffusion and reaction processes when solving the homogeneous equation is also presented. The two exponentially-fitted techniques presented in the paper are used on layer-adapted meshes which are piecewise uniform and concentrate grid points in the boundary layers, and their results are compared with those obtained with upwind methods in piecewise-uniform meshes. It is shown that standard techniques on piecewise-uniform meshes are less accurate than exponentially-fitted ones, and the accuracy of the latter may not improve by employing layer-adapted piecewise-uniform meshes due to the large change in the step size at the transition points. The paper also presents an exponentially-fitted method for singularly-perturbed, periodic, two-point boundary-value problems of ordinary differential equations.

Numerical analysis of a singularly perturbed nonlinear reaction–diffusion problem with multiple solutions

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 investigate the accuracy of solutions of a standard three-point difference scheme on layer-adapted meshes of Bakhvalov and Shishkin types. It is shown that one gets second-order convergence (with, in the case of the Shishkin mesh, a logarithmic factor) in the discrete maximum norm, uniformly in ɛ for ɛ ⩽ C N − 1 , where N is the number of mesh intervals. Numerical experiments are performed to support the theoretical results.

Layer-adapted meshes for one-dimensional reaction–convection–diffusion problems

Layer-adapted meshes for one-dimensional reaction–convection–diffusion problems

How accurate is the streamline-diffusion FEM inside characteristic (boundary and interior) layers?

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 characteristic-layer regions if anisotropic layer-adapted meshes are used. It is shown that the streamline-diffusion formulation may, in the maximum norm, imply only first-order accuracy in characteristic-layer regions. Numerical experiments are presented that support our theoretical predictions.

The convergence of a finite difference method on layer-adapted mesh for a singularly perturbed system

This paper is concerned with the numerical solution for singular perturbation system of two coupled ordinary differential equations with first and second orders and with initial and boundary conditions, respectively. Finite difference scheme on a special non-uniform mesh, whose solution converges pointwise independently of the singular perturbation parameter is constructed and analyzed. Numerical results supporting the theory are presented.

Layer-adapted meshes for one-dimensional reaction–convection–diffusion problems

We study convergence properties of an upwinded finite element method for the solution of linear one-dimensional reaction–convection–diffusion problems on arbitrary meshes. We derive conditions that are sufficient for (almost) first-order convergence in the L ∞ norm, uniformly in the diffusion parameter, of the method. These conditions are easy to check and enable one to immediately deduce the rate of convergence. The key ingredients of our analysis are sharp bounds on the W 1,1 norm of the discrete Green's function associated with the discretization.

A finite element analysis of a coupled system of singularly perturbed reaction–diffusion equations

We consider a system of two coupled reaction–diffusion equations. When the parameters multiplying the second-order derivatives in the equations are small, their solutions exhibit boundary layers. Moreover, when the parameters are of different magnitudes, two distinct but overlapping boundary layers are present. We study a finite element discretization on general layer-adapted meshes including the frequently studied Shishkin mesh and the Bakhvalov mesh. Supporting numerical results are presented.