Conditional ε-uniform boundedness of Galerkin projectors and convergence of an adaptive mesh method as applied to singularly perturbed boundary value problems

The Galerkin finite element method is applied to nonself-adjoint singularly perturbed boundary value problems on Shishkin meshes. The Galerkin projection method is used to obtain conditionally ε-uniform a priori error estimates and to prove the convergence of a sequence of meshes in the case of an unknown boundary layer edge.     

A finite element method for two–parameter singularly perturbed problems in 2D

We consider a singularly perturbed elliptic problem with two small parameters posed on the unit square. Based on a decomposition of the solution, we prove uniform convergence of a finite element method in an energy norm. The method uses piecewise bilinear functions on a layer-adapted Shishkin mesh. Numerical results confirm our theoretical analysis. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Superconvergence of finite element method for singularly perturbed convection–diffusion equations in 1D

In this article, a singularly perturbed convection–diffusion equation is solved by a linear finite element method on a Shishkin mesh. By means of an analysis exploiting symmetries in the convective term of the bilinear form, a new superconvergence rate, which improves the existing result, is obtained.     

A P0–P0 weak Galerkin finite element method for solving singularly perturbed reaction–diffusion problems

This paper investigates the lowest-order weak Galerkin finite element (WGFE) method for solving reaction–diffusion equations with singular perturbations in two and three space dimensions. The system of linear equations for the new scheme is positive definite, and one might readily get the well-posedness of the system. Our numerical experiments confirmed our error analysis that our WGFE method of the lowest order could deliver numerical approximations of the order O(h^1/2) and O(h) in H¹ and L² norms, respectively.     

Convergence of a finite element method on a Bakhvalov-type mesh for a singularly perturbed convection–diffusion equation in 2D

A finite element method of any order is applied on a Bakhvalov-type mesh to solve a singularly perturbed convection–diffusion equation in 2D, whose solution exhibits exponential boundary layers. A uniform convergence of (almost) optimal order is proved by means of a carefully defined interpolant.     

A posteriori error estimates of finite element method for parabolic problems

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Necessary conditions for ɛ-uniform convergence of finite difference schemes for parabolic equations with moving boundary layers

A grid approximation of the boundary value problem for a singularly perturbed parabolic reaction-diffusion equation is considered in a domain with the boundaries moving along the axis x in the positive direction. For small values of the parameter ? (this is the coefficient of the higher order derivatives of the equation, ? ∈ (0, 1]), a moving boundary layer appears in a neighborhood of the left lateral boundary S ₁ ^L . In the case of stationary boundary layers, the classical finite difference schemes on piece-wise uniform grids condensing in the layers converge ?-uniformly at a rate of O(N ^?1lnN + N ₀), where N and N ₀ define the number of mesh points in x and t. For the problem examined in this paper, the classical finite difference schemes based on uniform grids converge only under the condition N ^?1 + N ₀ ^?1 ? ?. It turns out that, in the class of difference schemes on rectangular grids that are condensed in a neighborhood of S ₁ ^L with respect to x and t, the convergence under the condition N ^?1 + N ₀ ^?1 ≤ ?^1/2 cannot be achieved. Examination of widths that are similar to Kolmogorov’s widths makes it possible to establish necessary and sufficient conditions for the ?-uniform convergence of approximations of the solution to the boundary value problem. These conditions are used to design a scheme that converges ?-uniformly at a rate of O(N ^?1lnN + N ₀).     

A quadratic finite element method for solving biharmonic problems in ℝ n

Summary A family of simplicial finite element methods having the simplest possible structure, is introduced to solve biharmonic problems in ⁿ ,n3, using the primal variable. The family is inspired in the MORLEY triangle for the two dimensional case, and in some sense this element can be viewed as its member corresponding to the valuen=2.     

An adaptive finite element method in reconstruction of coefficients in Maxwell’s equations from limited observations

We propose an adaptive finite element method for the solution of a coefficient inverse problem of simultaneous reconstruction of the dielectric permittivity and magnetic permeability functions in the Maxwell’s system using limited boundary observations of the electric field in 3D.     

Guaranteed a posteriori error estimates for nonconforming finite element approximations to a singularly perturbed reaction–diffusion problem

We derive guaranteed a posteriori error estimates for nonconforming finite element approximations to a singularly perturbed reaction–diffusion problem. First, an abstract a posteriori error bound is derived under a special equilibration condition. Based on conservative flux reconstruction, two error estimators are proposed and provide actual upper error bounds in the usual energy norm without unknown constants, one of which can be directly constructed without solving local Neumann problems and provide practical computable error bounds. The error estimators also provide local lower bounds but with the multiplicative constants dependent on the diffusion coefficient and mesh size, where the constants can be bounded for enough small mesh size comparable with the square root of the diffusion coefficient. By adding edge jumps with weights to the energy norm, two modified error estimators with additional edge tangential jumps are shown to be robust with respect to the diffusion coefficient and provide guaranteed upper bounds on the error in the modified norm. Finally, the performance of the estimators are illustrated by the numerical results.     

Novel fitted operator finite difference methods for singularly perturbed elliptic convection–diffusion problems in two dimensions

We consider a class of singularly perturbed elliptic problems posed on a unit square. These problems are solved by using fitted mesh methods by many researchers but no attempts are made to solve them using fitted operator methods, except our recent work on reaction–diffusion problems [J.B. Munyakazi and K.C. Patidar, Higher order numerical methods for singularly perturbed elliptic problems, Neural Parallel Sci. Comput. 18(1) (2010), pp. 75–88]. In this paper, we design two fitted operator finite difference methods (FOFDMs) for singularly perturbed convection–diffusion problems which possess solutions with exponential and parabolic boundary layers, respectively. We observe that both of these FOFDMs are ?-uniformly convergent. This fact contradicts the claim about singularly perturbed convection–diffusion problems [Miller et al. Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, Singapore, 1996] that ‘when parabolic boundary layers are present, …, it is not possible to design an ?-uniform FOFDM if the mesh is restricted to being a uniform mesh’. We confirm our theoretical findings through computational investigations and also found that we obtain better results than those of Linß and Stynes [Appl. Numer. Math. 31 (1999), pp. 255–270].     

L∞-boundedness of the finite element galerkin operator for parabolic problems

In this paper the heat equation with Dirichlet boundary conditions in N ≤ 3 space dimensions - serving as model problem of second order parabolic initial boundary value problems - is considered. We prove: The standard finite element method is uniformly bounded in L_∞ with respect to space and time if the underlying finite elements are at least cubics.     

A higher-order hybrid spline difference method on adaptive mesh for solving singularly perturbed parabolic reaction–diffusion problems with robin-boundary conditions

We propose a hybrid numerical scheme to discretize a class of singularly perturbed parabolic reaction–diffusion problems with robin-boundary conditions on an equidistributed grid. The hybrid difference scheme is developed by using a modified backward difference scheme in time, a combination of the cubic spline and exponential spline difference scheme in space. The proposed scheme uses a cubic spline difference scheme for the discretization of robin-boundary conditions. For the time discretization of the problem, we use the standard uniform mesh while a layer adapted equidistributed grid is generated for the spatial discretization. By equidistributing a curvature-based monitor function, the spatial adaptive grid is able to capture the presence of parabolic boundary layers without using any prior information about the solution. Parameter uniform error estimates are derived to illustrate an optimal convergence of first-order in time and second-order in space for the proposed discretization. The accuracy of the proposed scheme is confirmed by the numerical experiments that underpin the theoretical analysis.     

Uniform convergence of multigrid V-cycle on adaptively refined finite element meshes for second order elliptic problems

In this paper we prove the uniform convergence of the standard multigrid V-cycle algorithm with the Gauss-Seidel relaxation performed only on the new nodes and their "immediate" neighbors for discrete elliptic problems on the adaptively refined finite element meshes using the newest vertex bisection algorithm. The proof depends on sharp estimates on the relationship of local mesh sizes and a new stability estimate for the space decomposition based on the Scott-Zhang interpolation operator. Extensive numerical results are reported, which confirm the theoretical analysis.     

Limitations of Richardson’s extrapolation for a high order fitted mesh method for self-adjoint singularly perturbed problems

The aim of this paper is to investigate whether we can accelerate the order of convergence of existing high order methods to solve some singularly perturbed two-point BVPs. To this end, we consider a fitted mesh finite difference method of Patidar (Appl. Math. Comput., 188:720–733, 2007) applied on a mesh of Shishkin type for the solution of self-adjoint problem which is ε-uniformly convergent of order four. We attempted to increase the order of convergence by Richardson’s extrapolation and discovered that this well-known convergence acceleration technique has some limitations. We observe that even though this extrapolation technique improves the accuracy slightly, it does not increase the rate of convergence which is originally four for the underlying method for the problem above. Theoretical investigations are demonstrated by some numerical experiments.