A discontinuous mixed covolume method for elliptic problems

Abstract

a b s t r a c tWe develop a discontinuous mixed covolume method for elliptic problems on triangular meshes. An optimal error estimate for the approximation of velocity is obtained in a meshdependent norm. First-order L 2 -error estimates are derived for the approximations of both velocity and pressure.

Discontinuous Mixed Covolume Methods for Linear Parabolic Integrodifferential Problems

“…Differentiating each equation of (45) on and using (43), (44) we can prove this theorem in the same way as [15].…”

“…It was proved in [15] that the above formula has a unique solution and the error estimates in the following Theorem 7.…”

“…The discontinuous finite volume method in recent years was used to solve elliptic problems, Stokes problems, and parabolic problems in [12][13][14]. In [15] the discontinuous mixed covolume methods for elliptic problems were demonstrated by Yang and Jiang. Zhu and Jiang extended the discontinuous mixed covolume methods to parabolic problems in [16].…”

Discontinuous Mixed Covolume Methods for Parabolic Problems

“…Define a discontinuous mixed covolume elliptic projection by requiring that, finding , such that It was proved in [ 15 ] that ( 46 ) has a unique solution and the error estimates in Theorem 8 .…”

“…The discontinuous finite volume method was used for parabolic equations by Bi and Geng in [ 14 ]. In [ 15 ], Yang and Jiang extended a new discontinuous mixed covolume method for elliptic problems. In this paper, we consider the semidiscrete and the backward Euler fully discrete discontinuous mixed covolume methods for the second-order parabolic problems and derive the optimal order error estimates in the discontinuous H (div⁡) and first-order L 2 -error estimates in a mesh-dependent norm.…”