In this article we consider the singularly perturbed reaction-diffusion equation
where is a positive parameter. Assume that and are sufficiently smooth and
with a positive constant . Under these conditions on the data in problem (1.1), there exists a unique solution in for all . The solution to problem (1.1) typically exhibits boundary layers of width along all of in the singularly perturbed case of interest.
For singularly perturbed problems, it is popular to introduce layer-adapted meshes Linb:2010-Layer ; Roo1Sty2Tob3:2008-Robust ; Stynes:2005-Steady to fully resolve layers. Then uniform convergence with respect to singular perturbation parameter can be achieved for standard numerical methods. There are two kinds of layer-adapted meshes widely used in the literature, which are Bakhvalov-type mesh (B-mesh) and Shishkin-type mesh (S-mesh) (see Linb:2010-Layer ). There are a lot of research results on convergence theories of finite element methods on S-meshes; see Roo1Sty2Tob3:2008-Robust ; Franz:2008-Singularly ; Dur1Lom2Pri3:2013-Supercloseness ; Zhan1Liu2:2016-Analysis-CL ; Zhan1Liu2:2017-Supercloseness-EL ; Zhan1Styn2:2017-Supercloseness ; Liu1Sty2Zha3:2018-Supercloseness-CIP-EL and references therein.
Although B-meshes usually have better performances than S-meshes, there are very few articles on uniform convergence of finite element methods on the former. One main reason is that B-meshes have specific transition points between the fine and coarse parts, which are independent of mesh parameter. In the meantime, they bring great difficulties to convergence analysis. For example, Lagrange interpolant does not work well for B-meshes. Recently, Zhang and Liu Zhan1Liu2:2020-Optimal ; Zhan1Liu2:2021-Supercloseness proposed an variant of Lagrange interpolant for finite element methods on B-meshes in the case of convection-diffusion equations and succeeded to obtain a uniform convergence of optimal order.
For reaction-diffusion problems, they also proved optimal order of uniform convergence in the natural energy norm in Zhan1Liu2:2020-Convergence . However, energy norm is not strong enough to capture layers as the singular perturbation parameter tends to zero. Thus balanced norms, which are stronger than standard energy norms and characterize layers in an more appropriate way, were introduced in Lin1Styn2:2012-balanced for a mixed finite element method and Roos1Scho2:2015-Convergence for a finite element method. The authors Roos1Scho2:2015-Convergence introduced an
-projection to obtain desired estimations by-stability of the -projection on Shishkin mesh. To improve estimations in Roos1Scho2:2015-Convergence , the authors Fran1Roos2:2019-Error introduced a new interpolation, which consists a local weighted projection defined on the uniform part of S-meshes. Unfortunately, unlike S-meshes, there is little development on convergence theories in balanced norms on B-meshes.
In this manuscript we analyze convergence theories in the balanced norm introduced in Roos1Scho2:2015-Convergence for th () order finite element method on Bakhvalov-type rectangular meshes. For this purpose, we propose a novel interpolant according to the structures of B-meshes and layer functions. This interpolant consists of a local weighted projection defined on a proper mesh subdomain and the Lagrange interpolant. To prove the convergence of optimal order in the balanced norm, we must take into account the scales of the meshes and different stabilities of the projection. The optimal order convergence is also supported by our numerical experiments. Furthermore, we propose another novel interpolant operator, which consists of the local weighted projection operator, a vertices-edges-element operator and some corrections on the boundary. By careful derivations, we obtain a supercloseness result, which appears in the literature for the first time. Here “supercloseness” means that the convergence order for the error between some interpolation of the solution and the numerical solution in some norm is greater than the order for in the same norm.
The rest of the paper is organized as follows. In Section 2 we present a priori information of the solution to (1.1), then introduce Bakhvalov-type meshes, finite element methods and some preliminary results. In Section 3 we give a new interpolant and prove uniform convergence of optimal order in the balanced norm. Supercloseness result is given in Section 4 by means of another novel interpolant. In Section 5, numerical results illustrate our theoretical results.
Let . In this article, we will write for the inner product in , , , and for the standard norms in , , and the standard seminorm in , respectively. If , the subscript will be omitted from the above norm designations. Throughout the paper, all constants and are independent of and ; the constants are generic while subscripted constants are fixed.
2 Finite element method on Bakhvalov-type mesh
2.1 Regularity results
To construct layer-adapted meshes and analyze uniform convergence, we need a priori information of the solution to (1.1), such as pointwise estimations of the derivatives of the solution, the locations and widths of layers.
The solution of (1.1) can be decomposed as
where is the regular part, each is a boundary layer function and each is a corner layer function. For and , there exists a constant such that
and similarly for the remaining terms. Here denote by .
In the following analysis, we will denote by .
2.2 Bakhvalov-type meshes
Two Bakhvalov-type meshes will be discussed. Let be divisible by 4. The first Bakhvalov-type mesh is introduced in Roos-2006-Error and defined by
where will be defined later and , are used to ensure the continuity of at and . The second Bakhvalov-type mesh is introduced in Kopteva:1999-the ; Kopt1Save2:2011-Pointwise and its mesh generating function is
where will be specified later, with some positive constant independent of and , and are chosen so that is continuous at and . The original Bakhvalov mesh Bakhvalov:1969-Towards can be recovered from (2.6) by setting with .
Assume that in our analysis. In practice it is not a restriction.
For technical reasons, we also assume
Assume . Under Assumption 2 and (2.7), we have and for meshes (2.5) and (2.6). The location of and conditions imposed on and will simplify our later analysis without changing the essential difficulties in our analysis.
The mesh points are or for . By drawing lines parallel to the axis through mesh points , we obtain a Bakhvalov-type rectangular mesh with equidistant cells in the coarse region and anisotropic cells in the layer region . The triangulation is denoted by . Denote by for the element and by for a generic rectangular element, which dimensions are written as and . Define for .
In the following lemma, we collect some important properties possessed by Bakhvalov-type meshes, which are important for convergence analysis. The reader is referred to Zhan1Liu2:2020-Optimal for the detailed proof.
2.3 Finite element method
with . The natural energy norm derived from is
The bilinear form is coercive with respect to this energy norm, i.e.,
From the Lax-Milgram lemma, the weak formulation (2.14) has a unique solution.
Let denote th rectangular finite element. We introduce the finite element spaces
Clearly, . When we replace the infinite dimensional space with the finite dimensional space , we get the th order finite element method
Also, it is easy to very the coercivity
and the Galerkin orthogonality
Clearly, the balanced norm is stronger than the energy norm in the case of . Furthermore, the former is better suited to capture of layers. For example, for a typical layer function , and are of order and of order , respectively. These orders imply that the balanced norm is more appropriate to capture layers than the energy norm when . The reader is also referred to Roos1Scho2:2015-Convergence ; Con1Fra2Lud3etc:2018-Finite for discussions on these two norms.
3 Uniform convergence
For convergence analysis in the balanced norm, we will present an interpolation operator, which components will be introduced at first.
Set and . Introduce a weighted -projection as follows: for , find such that
where . Of course, one has the -stability
Denote by the Langrange interpolation operator from to . Furthermore, define by
where are the interpolation points of the Lagrange interpolation.
Recall . Then the interpolation used in convergence analysis is defined by
The following lemma provides some pointwise bounds for errors between the Lagrange interpolant and the projection.
For introduced in Assumption 1, one has
Inverse inequalities and the -stability of the -projection (3.1) yield
The error is split as follows:
In the following analysis, we give the estimations on each term in the right-hand side of (3.5).
Our arguments are based on the following splitting
From the definition of , we obtain
From (Zhan1Liu2:2020-Convergence, , Lemma 4), one has
To analyze , we need the following bounds
which could be derived from standard interpolation theories and Lemma 2. Then we obtain
Similar to (Zhan1Liu2:2020-Convergence, , Lemma 4), we have
and similarly have
Thus we obtain
In order to derive the supercloseness result in the balanced norm, we need another novel interpolant, which will be described in the following.
Instead of Langrange interpolant operator used in the previous section, we introduce an vertices-edges-element interpolation operator (see Lin1Yan2Zho3:1991-rectangle ; Styn1Tobi2:2008-Using ). This interpolant is used for superconvergence analysis of the diffusion term. First we define the interpolant operator on the reference element , whose vertices and edges are denoted by and respectively for . Let . The operator is determined by continuous linear functionals , which are defined by
From (Styn1Tobi2:2008-Using, , Lemma 3), the operator is uniquely determined. Then using the affine transformation to map from to an arbitrary , one obtains the corresponding interpolation operator . At last a continuous global interpolation operator is defined by setting
Besides, we denote by degree of freedom (DoF) of , which originates from the linear functional .
Recall and . The operator is defined by
If is the DoF of attached to , then it must be one of the following forms
In fact, is introduced for the continuity of at . Set . The operator is defined by