Convergence and superconvergence analysis of an anisotropic nonconforming finite element methods for singularly perturbed reaction-diffusion problems

The numerical approximation by a lower order anisotropic nonconforming finite element on appropriately graded meshes are considered for solving singular perturbation problems. The quasi-optimal order error estimates are proved in the ε-weighted H¹-norm valid uniformly, up to a logarithmic factor, in the singular perturbation parameter. By using the interpolation postprocessing technique, the global superconvergent error estimates in ε-weighted H¹-norm are obtained. Numerical experiments are given to demonstrate validity of our theoretical analysis.     

A-posteriori error analysis and adaptive finite element methods for singularly perturbed convection-diffusion equations

For singularly perturbed one-dimensional convection-diffusion equations, finite element approximations are constructed based on a so-called approximate symmetrization of the given unsymmetric problem. Local a-posteriori error estimates are established with respect to an appropriate energy norm where the bounds are proved to be realistic. The local bounds, called error indicators, provide a basis for a self-adaptive mesh refinement. For a model problem numerical results are presented showing that the adaptive method detects and resolves the boundary layer.     

Conditioning of finite difference schemes for a singularly perturbed convection-diffusion parabolic equation

In the case of the boundary value problem for a singularly perturbed convection-diffusion parabolic equation, conditioning of an ε-uniformly convergent finite difference scheme on a piecewise uniform grid is examined. Conditioning of a finite difference scheme on a uniform grid is also examined provided that this scheme is convergent. For the condition number of the scheme on a piecewise uniform grid, an ε-uniform bound O(δ ₁ ^?2 lnδ ₁ ^?1 + δ ₀ ^?1 ) is obtained, where δ₁ and δ₀ are the error components due to the approximation of the derivatives with respect to x and t, respectively. Thus, this scheme is ε-uniformly well-conditioned. For the condition number of the scheme on a uniform grid, we have the estimate O(ε^?1δ ₁ ^?2 + δ ₀ ^?1 ); this scheme is not ε-uniformly well-conditioned. In the case of the difference scheme on a uniform grid, there is an additional error due to perturbations of the grid solution; this error grows unboundedly as ε → 0, which reduces the accuracy of the grid solution (the number of correct significant digits in the grid solution is reduced). The condition numbers of the matrices of the schemes under examination are the same; both have an order of O(ε^?1δ ₁ ^?2 + δ ₀ ^?1 ). Neither the matrix of the ε-uniformly convergent scheme nor the matrix of the scheme on a uniform grid is ε-uniformly well-conditioned.     

Sequential and parallel domain decomposition methods for a singularly perturbed parabolic convection-diffusion equation

A boundary value problem for a singularly perturbed parabolic convection-diffusion equation is considered in a rectangular domain in x and t; the perturbation parameter ? multiplying the highest derivative takes arbitrary values in the half-open interval (0,1]. For the boundary value problem, we construct a scheme based on the method of lines in x passing through N ₀+1 points of the mesh with respect to t. To solve the problem on a set of intervals, we apply a domain decomposition method (on overlapping subdomains with the overlap width δ), which is a modification of the Schwarz method. For the continual schemes of the decomposition method, we study how sequential and parallel computations, the order of priority in which the subproblems are sequentially solved on the subdomains, and the value of the parameter ? (as well as the values of N ₀, δ) influence the convergence rate of the decomposition scheme (as N ₀ → ∞), and also computational costs for solving the scheme and time required for its solution (unless a prescribed tolerance is achieved). For convection-diffusion equations, in contrast to reaction-diffusion ones, the sequential scheme turns out to be more efficient than the parallel scheme.     

Computer difference scheme for a singularly perturbed convection-diffusion equation

The Dirichlet problem for a singularly perturbed ordinary differential convection-diffusion equation with a perturbation parameter ? (that takes arbitrary values from the half-open interval (0, 1]) is considered. For this problem, an approach to the construction of a numerical method based on a standard difference scheme on uniform meshes is developed in the case when the data of the grid problem include perturbations and additional perturbations are introduced in the course of the computations on a computer. In the absence of perturbations, the standard difference scheme converges at an \(\mathcal{O}\) (δ _st ) rate, where δ _st = (? + N ^?1)^?1 N ^?1 and N + 1 is the number of grid nodes; the scheme is not ?-uniformly well conditioned or stable to perturbations of the data. Even if the convergence of the standard scheme is theoretically proved, the actual accuracy of the computed solution in the presence of perturbations degrades with decreasing ? down to its complete loss for small ? (namely, for ? = \(\mathcal{O}\) (δ^?2max _i,j |δa _i ^j | + δ^?1 max _{i, j} |δb _i ^j |), where δ = δ _st and δa _i ^j , δb _i ^j are the perturbations in the coefficients multiplying the second and first derivatives). For the boundary value problem, we construct a computer difference scheme, i.e., a computing system that consists of a standard scheme on a uniform mesh in the presence of controlled perturbations in the grid problem data and a hypothetical computer with controlled computer perturbations. The conditions on admissible perturbations in the grid problem data and on admissible computer perturbations are obtained under which the computer difference scheme converges in the maximum norm for ? ∈ (0, 1] at the same rate as the standard scheme in the absence of perturbations.     

A Shishkin mesh for a singularly perturbed Riccati equation

In this paper a singularly perturbed Riccati initial value problem is examined. Parameter explicit bounds on the solution and its derivatives are given. A numerical method composed of an implicit difference operator and a piecewise-uniform Shishkin mesh is constructed. A theoretical parameter independently bound on the errors in the numerical approximations is established. Numerical results are presented which are in agreement with the theoretical error bound.     

A posteriori error estimate for finite volume approximations to singularly perturbed nonlinear convection-diffusion equations

Summary. This paper is devoted to the study of a posteriori and a priori error estimates for the scalar nonlinear convection diffusion equation . The estimates for the error between the exact solution and an upwind finite volume approximation to the solution are derived in the -norm in the situation, where the diffusion parameter is smaller or comparable to the mesh size. Numerical experiments underline the theoretical results. Received February 25, 1999 / Revised version received July 6, 1999 / Published online August 2, 2000     

Uniform mesh approximation to nonsmooth solutions of a singularly perturbed convection-diffusion equation in a rectangle

Here nonsmooth solutions of a differential equation are treated as solutions for which the compatibility conditions are not required to hold at the corner points of the domain and hence corner singularities can occur. In the present paper, we drop the compatibility conditions at three of the four vertices of a rectangle. At the remaining vertex, from which a characteristic (inclined) of the reduced equation issues, we impose compatibility conditions providing the C ^3,λ -smoothness of the desired solution in a neighborhood of that vertex as well as additional conditions leading to the smoothness of solutions of the reduced equation occurring in the regular component of the solution of the considered problem. Under our assumptions and for a sufficient smoothness of the coefficients of the equation and its right-hand side, we show that the classical five-point upwind approximation on a Shishkin piecewise uniform mesh preserves the accuracy specific for the smooth case; i.e., the mesh solution uniformly (with respect to a small parameter) converges in the L _∞ ^h -norm to the exact solution at the rate O(N ⁻¹ ln² N), where N is the number of mesh nodes in each of the coordinate directions.     

Convergence and superconvergence analysis of an anisotropic nonconforming finite element methods for semisingularly perturbed reaction–diffusion problems

The numerical approximation by a lower‐order anisotropic nonconforming finite element on appropriately graded meshes are considered for solving semisingular perturbation problems. The quasi‐optimal‐order error estimates are proved in the ε‐weighted H¹‐norm valid uniformly, up to a logarithmic factor, in the singular perturbation parameter. By using the interpolation postprocessing technique, the global superconvergent error estimates in ε‐weighted H¹‐norm are obtained. Numerical experiments are given to demonstrate validity of our theoretical analysis. Copyright © 2007 John Wiley & Sons, Ltd.     

Analysis of a combined barycentric finite volume—nonconforming finite element method for nonlinear convection-diffusion problems

We present the convergence analysis of an efficient numerical method for the solution of an initial-boundary value problem for a scalar nonlinear conservation law equation with a diffusion term. Nonlinear convective terms are approximated with the aid of a monotone finite volume scheme considered over the finite volume barycentric mesh, whereas the diffusion term is discretized by piecewise linear nonconforming triangular finite elements. Under the assumption that the triangulations are of weakly acute type, with the aid of the discrete maximum principle, a priori estimates and some compactness arguments based on the use of the Fourier transform with respect to time, the convergence of the approximate solutions to the exact solution is proved, provided the mesh size tends to zero.     

A higher order finite element method with modified graded mesh for singularly perturbed two-parameter problems

This paper presents a modified graded mesh for singularly perturbed two-parameter problems. The mesh is generated recursively using Newton's algorithm and some implicitly defined function. The problem is solved numerically using the finite element method based on higher order polynomials of degree p≥1. We prove parameter uniform convergence of optimal order in ε-weighted energy norm. A test example is taken to compare the proposed graded mesh with others found in the literature.     

A higher order richardson scheme for a singularly perturbed semilinear elliptic convection-diffusion equation

The Dirichlet problem on a vertical strip is examined for a singularly perturbed semilinear elliptic convection-diffusion equation. For this problem, the basic nonlinear difference scheme based on the classical approximations on piecewise uniform grids condensing in the vicinity of boundary layers converges ɛ-uniformly with an order at most almost one. The Richardson technique is used to construct a nonlinear scheme that converges ɛ-uniformly with an improved order, namely, at the rate O(N ₁⁻²ln² N ₁ + N ₂⁻²), where N ₁ + 1 and N ₂ + 1 are the number of grid nodes along the x ₁-axis and per unit interval of the x ₂-axis, respectively. This nonlinear basic scheme underlies the linearized iterative scheme, in which the nonlinear term is calculated using the values of the sought function found at the preceding iteration step. The latter scheme is used to construct a linearized iterative Richardson scheme converging ɛ-uniformly with an improved order. Both the basic and improved iterative schemes converge ɛ-uniformly at the rate of a geometric progression as the number of iteration steps grows. The upper and lower solutions to the iterative Richardson schemes are used as indicators, which makes it possible to determine the iteration step at which the same ɛ-uniform accuracy is attained as that of the non-iterative nonlinear Richardson scheme. It is shown that no Richardson schemes exist for the convection-diffusion boundary value problem converging ɛ-uniformly with an order greater than two. Principles are discussed on which the construction of schemes of order greater than two can be based.     

A finite difference scheme of improved accuracy on a priori adapted grids for a singularly perturbed parabolic convection-diffusion equation

In the case of the Dirichlet problem for a singularly perturbed parabolic convection-diffusion equation with a small parameter ɛ multiplying the higher order derivative, a finite difference scheme of improved order of accuracy that converges almost ɛ-uniformly (that is, the convergence rate of this scheme weakly depends on ɛ) is constructed. When ɛ is not very small, this scheme converges with an order of accuracy close to two. For the construction of the scheme, we use the classical monotone (of the first order of accuracy) approximations of the differential equation on a priori adapted locally uniform grids that are uniform in the domains where the solution is improved. The boundaries of such domains are determined using a majorant of the singular component of the grid solution. The accuracy of the scheme is improved using the Richardson technique based on two embedded grids. The resulting scheme converges at the rate of O((ɛ⁻¹ N ^−K ln² N)² + N ⁻²ln⁴ N + N ₀⁻²) as N, N ₀ → ∞, where N and N ₀ determine the number of points in the meshes in x and in t, respectively, and K is a prescribed number of iteration steps used to improve the grid solution. Outside the σ-neighborhood of the lateral boundary near which the boundary layer arises, the scheme converges with the second order in t and with the second order up to a logarithmic factor in x; here, σ = O(N ^{−(K − 1)}ln² N). The almost ɛ-uniformly convergent finite difference scheme converges with the defect of ɛ-uniform convergence ν, namely, under the condition N ⁻¹ ≪ ɛ^ν, where ν determining the required number of iteration steps K (K = K(ν)) can be chosen sufficiently small in the interval (0, 1]. When ɛ⁻¹ = O(N ^{K − 1}), the scheme converges at the rate of O(N ⁻²ln⁴ N + N ₀⁻²).     

Conditioning and stability of finite difference schemes on uniform meshes for a singularly perturbed parabolic convection-diffusion equation

For a singularly perturbed parabolic convection-diffusion equation, the conditioning and stability of finite difference schemes on uniform meshes are analyzed. It is shown that a convergent standard monotone finite difference scheme on a uniform mesh is not ?-uniformly well conditioned or ?-uniformly stable to perturbations of the data of the grid problem (here, ? is a perturbation parameter, ? ∈ (0, 1]). An alternative finite difference scheme is proposed, namely, a scheme in which the discrete solution is decomposed into regular and singular components that solve grid subproblems considered on uniform meshes. It is shown that this solution decomposition scheme converges ?-uniformly in the maximum norm at an O(N ^?1lnN + N ₀ ^?1 ) rate, where N + 1 and N ₀ + 1 are the numbers of grid nodes in x and t, respectively. This scheme is ?-uniformly well conditioned and ?-uniformly stable to perturbations of the data of the grid problem. The condition number of the solution decomposition scheme is of order O(δ^?2lnδ^?1 + δ ₀ ^?1 ); i.e., up to a logarithmic factor, it is the same as that of a classical scheme on uniform meshes in the case of a regular problem. Here, δ = N ^?1lnN and δ₀ = N ₀ ^?1 are the accuracies of the discrete solution in x and t, respectively.     

Stability and accuracy of adapted finite element methods for singularly perturbed problems

The stability and accuracy of a standard finite element method (FEM) and a new streamline diffusion finite element method (SDFEM) are studied in this paper for a one dimensional singularly perturbed connvection-diffusion problem discretized on arbitrary grids. Both schemes are proven to produce stable and accurate approximations provided that the underlying grid is properly adapted to capture the singularity (often in the form of boundary layers) of the solution. Surprisingly the accuracy of the standard FEM is shown to depend crucially on the uniformity of the grid away from the singularity. In other words, the accuracy of the adapted approximation is very sensitive to the perturbation of grid points in the region where the solution is smooth but, in contrast, it is robust with respect to perturbation of properly adapted grid inside the boundary layer. Motivated by this discovery, a new SDFEM is developed based on a special choice of the stabilization bubble function. The new method is shown to have an optimal maximum norm stability and approximation property in the sense that where u _N is the SDFEM approximation in linear finite element space V ^N of the exact solution u. Finally several optimal convergence results for the standard FEM and the new SDFEM are obtained and an open question about the optimal choice of the monitor function for the moving grid method is answered. This work was supported in part by NSF DMS-0209497 and NSF DMS-0215392 and the Changjiang Professorship through Peking University.     

A method of asymptotic constructions with improved accuracy for a quasilinear singularly perturbed parabolic convection-diffusion equation

The Dirichlet problem on an interval for quasilinear singularly perturbed parabolic convection-diffusion equation is considered. The higher order derivative of the equation is multiplied by a parameter ε that takes any values from the half-open interval (0, 1]. For this type of linear problems, the order of the ε-uniform convergence (with respect to x and t) for the well-known schemes is not higher than unity (in the maximum norm). For the boundary value problem under consideration, grid approximations are constructed that converge ε-uniformly at the rate of O(N ^?2ln² N + N ^?2 ₀), where N + 1 and N ₀ + 1 are the numbers of the mesh points with respect to x and t, respectively. On the x axis, piecewise uniform meshes that condense in the boundary layer are used. If the parameter value is small compared to the effective step of the spatial grid, the domain decomposition method is used, which is motivated by “asymptotic constructions.” Monotone approximations of “auxiliary” subproblems describing the main terms of the asymptotic expansion of the solution outside a neighborhood of the boundary layer neighborhood are used. In the neighborhood of the boundary layer (of the width O(ε ln N)) the first derivative with respect to x is approximated by the central difference derivative. These subproblems are successively solved in the subdomains on uniform grids. If the parameter values are not sufficiently small (compared to the effective step of the mesh with respect to x), the classical implicit difference schemes approximating the first derivative with respect to x by the central difference derivative are applied. To improve the accuracy in t, the defect correction technique is used. Notice that the calculation of the solution of the constructed difference scheme (the scheme based on the method of asymptotic constructions) can be considerably simplified for sufficiently small values of the parameter ε.     

A finite element method for a singularly perturbed boundary value problem

Summary We examine the problem:u+a(x)u–b(x)u=f(x) for 0<x<1,a(x)>0,b(x)>, ² = 4>0,a, b andf inC ² [0, 1], in (0, 1],u(0) andu(1) given. Using finite elements and a discretized Green's function, we show that the El-Mistikawy and Werle difference scheme on an equidistant mesh of widthh is uniformly second order accurate for this problem (i.e., the nodal errors are bounded byCh ², whereC is independent ofh and ). With a natural choice of trial functions, uniform first order accuracy is obtained in theL (0, 1) norm. On choosing piecewise linear trial functions (hat functions), uniform first order accuracy is obtained in theL ¹ (0, 1) norm.     

A uniform finite element method for a conservative singularly perturbed problem

We examine the problem ϵ(p(x)u′) + (q(x)u)′ − r(x)u = f(x) for 0 < x < 1, p>0, q>0, r ⩾ 0; p, q, r and f in C²[0, 1], ϵ in (0, 1], u(0) and u(1) given. Existence of a unique solution u and bounds on u and its derivatives are obtained. Using finite elements on an equidistant mesh of width h we generate a tridiagonal difference scheme which is shown to be uniformly second order accurate for this problem (i.e., the nodal errors are bounded by Ch², where C is independent of h and ϵ). With a natural choice of trial functions, uniform first order accuracy is obtained in the L^∞(0, 1) norm. Using trial functions which interpolate linearly between the nodal values generated by the difference scheme gives uniform first order accuracy in the L¹(0, 1) norm.