Mixed covolume method for parabolic problems on triangular grids☆
Introduction
Consider the following parabolic equation:where is a bounded polygonal domain in with the boundary , and is the outward unit normal vector on . The coefficient K(x) is a symmetric and uniformly positive-definite matrix, i.e. there exist two positive constants and such that . Furthermore, we assume that the matrix is Lipschitz.
Let us introduce a new variable and rewrite the problem (1.1) as the system of first order partial differential equationswhere . This system can be interpreted as modeling a compressible single phase flow in a reservoir ignoring gravitational effects. The variable p represents the pressure or head and u represents the Darcy velocity. The first equation represents the Darcy law, and the second represents conservation of mass with f standing for a source or sink term [16].
Now let us introduce the function spacesThen the associated weak formulation for (1.1) is to find such thatwhere denotes the standard inner product in or .
The purpose of this paper is to study the mixed covolume method for parabolic problems (1.1). Mixed covolume method was first proposed by Russell [1]. The basic technique of this method was to relate the Petro-Galerkin scheme to a standard finite element Galerkin or mixed method through an introduction of the transfer operator that maps the trial function space into the test function space. This method not only preserves the simplicity of finite difference and the high accuracy of finite element but maintains the mass conservation law, which is very important to fluid and under-ground fluid computations. The numerical experiment on a variety of test problems was very promising [13], [14]. The optimal convergence of the mixed covolume method for linear elliptic problems on triangular grids was given by Chou et al. [2], and they also extended it to rectangular grids and quadrilateral grids [3], [4]. Many equations based on this method have been developed and analyzed on rectangular grids [5], [6]. However, the analysis on triangular grids are relatively few. The goal of this article is to give the error estimates on triangular grids. We give the optimal convergence for the velocity and pressure in -norm and also give the quasi-optimal estimates for the pressure in -norm.
Throughout this paper, C will denote a generic constant independent of the discretization parameters and may take on different values in different appearances. We also adopt the standard definitions and notations of Sobolev spaces and their full norms and semi-norms as in [10], [15].
Section snippets
Mixed covolume formulation
In order to describe the mixed covolume method for the problem (1.1), we first construct the partition of the domain . As in [2], let be a quasi-uniform (regular) triangulation of the domain , see [10], [15], where stands for the triangle whose barycenter is B (cf. Fig. 1), and , is the diameter of the triangle . We denote by those nodes belonging to the interior of and those on the boundary.
Based on the partition , we select the lowest
Lemmas
In this section we give some properties of the operator which will be of crucial importance in establishing error estimates for (2.11). Lemma 3.1 For any function , the following holds:Define . Lemma 3.2 There exists a positive constant independent of h such that Proof Let and . Thensee Chou et al. [2]
Generalized mixed covolume elliptic projection
Before proceeding to the error analysis, we introduce a generalized mixed covolume elliptic projection associated with (1.3). Define a map such thatLet . By (2.4), (4.1) can be rewritten aswhere
The error estimates of continuous-in-times
Subtracting (1.3) from (2.11), we obtain the error equationsConsidering the mixed covolume elliptic projection introduced in (4.1), the error equations can be rewritten asLet . Then (5.2) becomes
The error estimation of backward Euler
The backward Euler of the mixed covolume formulation for the problem (1.1): to find , such thatSubtracting (1.3) from (6.1) and considering the mixed covolume elliptic projection introduced in (4.1), we can obtain the error equationsBy the notes in Section 5,
Acknowledgements
The authors are very grateful to referees for some useful and helpful suggestions.
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This work was supported by National Nature Science of China under grant no. 10271068 and National Nature Science of Shandong province under grant nos. Y2007A14, Z2006A02.