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 (seeMil1Rio2Shi3:2012-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.
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 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
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.
We collect some bounds of the layer function and the function on the Bakhvalov meshes in the following lemma.
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 .
where we have used (19) with and . For we have
Now we are ready to analyze . First we decompose into the following two parts
4 Uniform convergence
In the following we will analyze the terms in the right-hand side of (37). Hölder inequalities yield
We put the arguments for in the following lemma.
According to (25), the term is separated into three parts as follows:
From Hölder inequalities and inverse inequalities, one has
where (28) has been used.
Now we analyze the term . Note on and one has
Now we are in a position to present the main result.