Abstract
In this work, we systematically analyze a stabilized finite volume method for the Poisson equation. On stating the convergence of this method, optimal error estimates in different norms are obtained by establishing the adequate connections between the finite element and finite volume methods. Furthermore, some super-convergence results are established by using L 2 -projection method on a coarse mesh based on some regularity assumptions for Poisson equation. Finally, some numerical experiments are presented to confirm the theoretical findings.
Highlights
Finite volume method (FVM) as one of important numerical discretization techniques has been widely employed to solve the fluid dynamics problems [8]
The basic idea of FVM is to approximate discrete fluxes of a partial differential equation using a finite element procedure based on volumes or control volumes, so FVM is known as covolume methods, or box methods [1], marker and cell methods [5], generalized difference methods [17]
−6 −4 −3.8 −3.6 −3.4 −3.2 −3 −2.8 −2.6 −2.4 −2.2 log(h). From those Tables, we can see that the relative errors of pressure obtained by L2 projection method are better than these in Table 2, and the convergence orders confirm the results of Corollary 1, see Figure 1 for details
Summary
Finite volume method (FVM) as one of important numerical discretization techniques has been widely employed to solve the fluid dynamics problems [8]. The mixed finite element method has became a popular method for solving the partial differential equations arising in solid and fluid mechanics [4]. We provide a systematical finite volume analysis of the velocity projection stabilization method for Poisson equation. Different from [18], this paper considers the mixed finite volume method for Poisson equations. We provide the stability and optimal error estimates of numerical solution, and present some meaningful superconvergence results by using the L2-projections between different finite element spaces.
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