The authors in presented and analyzed the semidiscrete and full discrete symmetric finite volume schemes for a class of parabolic problems. Recently, the authors in studied FVE method for general self-adjoint elliptic problems. This important property, combined with adequate accuracy and ease of implementation, has attracted more people to do research in this field. The most important property of FVE method is that it can preserve the conservation laws (mass, momentum, and heat flux) on each control volume. We refer to the monographs for general presentation of these methods. Usually, the former one is called finite volume method, marker and cell (MAC) method, or cell-centered method, and the latter one is called finite volume element method (FVE), covolume method, or vertex-centered method.
The method can be formulated in the finite difference framework or in the Petrov-Galerkin framework. It has been widely used in several engineering fields, such as fluid mechanics, heat and mass transfer, and petroleum engineering. Problem ( 1) occurs in nonlocal reactive flows in porous media, viscoelasticity, and heat conduction through materials with memory.įinite volume method is an important numerical tool for solving partial differential equations. Here, a non-self-adjoint second-order strongly elliptic, and, an arbitrary second-order linear partial differential operator, both with coefficients depending smoothly on and, and are known functions, which are assumed to be smooth and satisfy certain compatibility conditions for and, so that ( 1) has a unique solution in certain Sobolev space. Where is a bounded domain in, , with smooth boundary, and. The main purpose of this paper is to study semidiscrete and full discrete finite volume element method (FVE) for parabolic integrodifferential equation of the form Based on the Crank-Nicolson method, a time discretization scheme is discussed and related error estimates are derived.
In this paper, we also study the lumped mass modification. For the spatially discrete scheme, optimal-order error estimates in, , and, norms for are obtained. Finite volume element schemes for non-self-adjoint parabolic integrodifferential equations are derived and stated.
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