Optimal order of uniform convergence for finite element method on Bakhvalov-type meshes

by   Jin Zhang, et al.

We propose a new analysis of convergence for a kth order (k> 1) finite element method, which is applied on Bakhvalov-type meshes to a singularly perturbed two-point boundary value problem. A novel interpolant is introduced, which has a simple structure and is easy to generalize. By means of this interpolant, we prove an optimal order of uniform convergence with respect to the perturbation parameter. Numerical experiments illustrate these theoretical results.



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1 Introduction

We consider the two-point boundary value problem


where is a positive parameter, , and are sufficiently smooth functions such that on and


with some constants and . The condition (2) ensures that the boundary value problem has a unique solution. In the cases of interest the diffusion parameter can be arbitrarily small and satisfies . Thus this problem is singuarly perturbed and its solution typically features a boundary layer of width at (see Roo1Sty2Tob3:2008-Robust ).

Solutions to singularly perturbed problems are characterized by the presence of boundary or interior layers, where solutions change rapidly. Numerical solutions of these problems are of significant mathematical interest. Classical numerical methods are often inappropriate, because in practice it is very unlikely that layers are fully resolved by common meshes. Hence specialised numerical methods are designed to compute accurate approximate solutions in an efficient way. For example, standard numerical methods on layer-adapted meshes, which are fine in layer regions and standard outside, are commonly used; see Roo1Sty2Tob3:2008-Robust ; Mil1Rio2Shi3:2012-Fitted and many references therein. On these meshes, classical numerical methods are uniformly convergent with respect to the singular perturbation parameter; see Linb:2010-Layer . Among them, there are two kinds of popular grids: Bakhvalov-type meshes (B-meshes) and Shishkin-type meshes (S-meshes); see Linb:2010-Layer .

The accuracy of finite difference methods on these locally refined meshes has been extensively studied and sharp error estimations have been derived (see

Mil1Rio2Shi3:2012-Fitted ; Kopteva:1999-the ; Linb:2010-Layer ). For instance, in Linb:2010-Layer the author presented convergence rates of and for a first-order upwind difference scheme on Bakhvalov grid Bakhvalov:1969-Towards and Shishkin grid Shishkin:1990-Grid , respectively, where is the number of mesh intervals in each coordinate direction. Usually, the performance of B-meshes is superior to that of S-meshes. This advantage is more and more obvious when higher-order schemes are used. Besides, the width of the mesh subdomain used to resolve the layer is for B-meshes and for S-meshes. The former is independent of the mesh parameter and this property will be important under certain circumstances.

For finite element methods, the development of numerical theories on B-meshes is completely different from one on S-meshes. On standard Shishkin meshes Stynes and O’Riordan Styn1ORior2:1997-uniformly derived a sharp uniform convergence in the energy norm for finite element method. Henceforward numerous articles deal with uniform convergence of finite element methods on S-meshes; see e.g. Roo1Sty2Tob3:2008-Robust ; Roos1Styn2:2015-Some ; Zha1Liu2Yan3:2016-Optimal-EL ; Zhan1Styn2:2017-Supercloseness ; Liu1Sty2Zha3:2018-Supercloseness-CIP-EL ; Teo1Brd2Fra3etc:2018-SDFEM ; Russ1Styn2:2019-Balanced-norm and the references therein. However, it is still open for the optimal uniform convergence of finite element methods on B-meshes ( see (Roos1Styn2:2015-Some, , Question 4.1) for more details).

This dilemma arises from the fact that the standard Lagrange interpolant does not work for uniform convergence of finite element methods on B-meshes. More specifically, the Lagrange interpolant cannot provide enough stability in norm on a special mesh interval, which lies in the fine part and is adjacent to the coarse part of B-meshes. In Roos-2006-Error and Brda1Zari2:2016-singularly a quasi-interpolant is used and provides enough stability for the optimal uniform convergence. Unfortunately, in both articles the analysis is limited to one dimension and linear finite element. It is hard to extend the analysis to higher dimensions or higher-order finite elements for singularly perturbed problems.

In this contribution we will study the optimal uniform convergence of a th order () finite element method on Bakhvalov-type meshes. A novel interpolant is constructed by redefining the standard Lagrange interpolant to the solution. This interpolant has a simple structure and it can also be applied to higher-dimensional problems in a straightforward way. By means of this novel function, we prove the optimal order of uniform convergence in a standard way.

The rest of the paper is organized as follows. In Section 2 we describe our regularity on the solution to (1), introduce two Bakhvalov-type meshes and define the finite element method. Some preliminary results for the subsequent analysis are also derived in this section. In Section 3 we construct and analyze an interpolant for the uniform convergence on B-meshes. In Section 4 uniform convergence is obtained by means of the interpolant and careful derivations of the convective term in the bilinear form. In Section 5, numerical results illustrate our theoretical bounds.

We use the standard Sobolev spaces , , for nonnegative integers and . Here is any measurable subset of . We denote by and the semi-norms and the norms in , respectively. On , and are the usual Sobolev semi-norm and norm. Denote by the norms in the Lebesgue spaces . We use the notation and for the -inner product and the -norm, respectively. When we drop the subscript from the notation for simplicity. Throughout the article, all constants and are independent of and the mesh parameter ; unsubscripted constants are generic and may take different values in different formulas while subscripted constants are fixed.

2 Regularity, Bakhvalov mesh and finite element method

2.1 Regularity of the solution

Information about higher-order derivatives of the solution of (1) are usually needed by uniform convergence of finite element methods. Such estimations appeared in (Roo1Sty2Tob3:2008-Robust, , Lemma 1.9) and are reproduced in the following lemma.

Lemma 1.

Let be some positive integer. Assume that (2) holds true and are sufficiently smooth. The solution of (1) can be decomposed into


where the smooth part and the layer part satisfy and , respectively. Furthermore, one has


Note depends on the regularity of the coefficients, in particular (4) holds for any if .

2.2 Bakhvalov mesh

Bakhvalov mesh first appeared in Bakhvalov:1969-Towards and is constructed according to layer functions like in Lemma 1. Its mesh generating function is piecewise and belongs to . Its breakpoint, which separates the mesh generating function, must be solved by a nonlinear equation and usually is not explicitly known ( see (Roo1Sty2Tob3:2008-Robust, , Part I §2.4.1)).

In this article, we focus on two Bakhvalov-type meshes introduced in Roos-2006-Error and Kopteva:1999-the ; Kopt1Save2:2011-Pointwise . Their breakpoints are known already, and both mesh generating functions do not belong to any longer. In Roos-2006-Error the Bakhvalov mesh is defined by


where with some positive integer and is used to ensure the continuity of at . The mesh generating function in Kopteva:1999-the ; Kopt1Save2:2011-Pointwise is defined by


where , with some positive constant independent of and , is chosen so that is continuous at . The original Bakhvalov mesh can be recovered from (6) by setting , where


For technical reasons, we assume and therefore . We also assume that in our analysis, as is generally the case in practice. If , one sets , which generate uniform meshes.

Assume that is a positive integer and define the mesh points or for . For both Bakhvalov meshes one usually has . Denote an arbitrary subinterval by , its length by and a generic subinterval by .

2.3 The finite element method

The weak form of problem (1) is to find such that


where . Note that the variational formulation (8) has a unique solution by means of the Lax-Milgram lemma.

Define the finite element space on the Bakhvalov meshes

The finite element method for (8) reads as


The natural norm associated with is defined by

Using (2), it is easy to see that one has the coercivity


with . It follows that is well defined by (9) (see Bren1Scot2:2008-mathematical and references therein).

2.4 Preliminary results of Bakhvalov meshes

In this subsection, we present some important properties of the Bakhvalov meshes and the layer function , which are necessary for our uniform convergence.

We present some properties about the step sizes of Bakhavlov meshes as follows.

Lemma 2.

For Bakhvalov mesh (5), one has


On Bakhvalov mesh (6), bounds analogous to (11)–(14) also hold.


We just consider Bakhvalov mesh (5) and the other mesh can be similarly analyzed. Recalling that and the Bakhvalov mesh separates into uniform subintervals, one obtains (14). For , one has



From (15), we can prove (11), (12) and (13) easily.

We collect some bounds of the layer function and the function on the Bakhvalov meshes in the following lemma.

Lemma 3.

On Bakhvalov meshes (5) and (6), one has


For and , we have


We just consider Bakhvalov mesh (5) and the mesh (6) can be similarly analyzed.

Recalling , we prove (16), (17) and (18) directly from (4).

Let . From (5) one has

and for


From (15) and (20), we have

where and for

Thus (19) is proved.

3 Interpolation operator and interpolation errors

Now a new interpolation operator is introduced, which is used for our uniform convergence. Set for and . For any its Lagrange interpolant on each Bakhvalov mesh is defined by

where , is the piecewise th order Lagrange basis function satisfying the well-known delta properties associated with the nodes and , respectively. For the solution to (1), recall (3) in Lemma 1 and define the interpolant by


where is the Lagrange interpolant to and




and clearly we have


Interpolation theories in Sobolev spaces (Ciarlet:2002-finite, , Theorem 3.1.4) tell us that


for all , where , and .

Lemma 4.

On Bakhvalov meshes (5) and (6), one has


where is defined in (23).


We just consider Bakhvalov mesh (5) and mesh (6) can be similarly analyzed.

From (26) and (4), for one has


where we have used (19) with and . For we have


Collecting (32), (33) and noting , we prove . Lemma 2, (26) and (4) yield . From (3) we prove (27). The bound (28) can be easily obtained from (27) and Hölder inequalities.

From (4) and direct calculations one can easily prove (29).

Now we are ready to analyze . First we decompose into the following two parts


From (26), (4), (19) with and , we have


From a triangle inequality, (13), (14), (17), (18), inverse inequality (Ciarlet:2002-finite, , Theorem 3.2.6) and (29), one has


Substituting (35), (36) into (34) and recalling and , we obtain

and prove from (28). From (26) and Lemma 2, one can easily prove . A triangle inequality yields . Thus (30) is proved.

Now we consider (31). Direct calculations yield

where we have used (23), (16), (12) and (13). ∎

4 Uniform convergence

Introduce . From (10), the Galerkin orthogonality, (3), (21), (24) and integration by parts for , one has


In the following we will analyze the terms in the right-hand side of (37). Hölder inequalities yield


where (30) and (31) have been used. From (26) and (4), one has and from (28) and (31). Consequently we obtain


We put the arguments for in the following lemma.

Lemma 5.

Let the mesh be either the Bakhvalov mesh (5) or the Bakhvalov mesh (6). Let be defined in (22). Then one has


According to (25), the term is separated into three parts as follows:


From Hölder inequalities, (26), (19) with and , we obtain


where (11) and (12) have been used.

From Hölder inequalities and inverse inequalities, one has


where (28) has been used.

Now we analyze the term . Note on and one has


where (22), Hölder inequalities, (26), (19) with and , (12) have been used. On , we have from (22) and


where Hölder inequalities, (16), (17) and (13) have been used. From (44) and (45) we prove


where and have been used. Substituting (42), (43) and (46) into (41), we are done. ∎

Now we are in a position to present the main result.

Theorem 1.

Let the mesh be either Bakhvalov mesh (5) or Bakhvalov mesh (6) with . Let and be the solutions of (1) and (9), respectively. Then one has


Substituting (38), (39) and (40) into (37), we obtain . From (24) and (31) we have . From a triangle inequality and (30), one has