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Finite volume method for degenerate diffusion problems

  • 35-Xx Partial Differential Equations
  • Mathematics


Finite volume method for degenerate diffusion problems Norikazu SAITO Graduate School of Mathematical Sciences The University of Tokyo 3-8-1 Komaba, Meguro, Tokyo, 153-8914 Japan [email protected] 1 Introduction The finite volume method (FVM) is a discretization method based on local con- servation properties of equations so that it is well suited for PDEs of conservation laws. Although the range of application seems to be smaller than that of the finite element method (FEM), FVM has its own advantages. For example, FVM natu- rally satisfies the discrete maximum principle, if it is applied to a linear diffusion problem. We recall that the discrete maximum principle in FEM holds only when some shape conditions on the triangulation are satisfied, and such a restriction often causes some difficulties. In this paper, we shall reveal another advantage of FVM through the degenerate diffusion problems and the nonlinear semigroup theory. The purpose of this paper is to report some operator theoretical properties of FVM applied to a degenerate elliptic equation of the form u−λ∆ f (u) = 1 for λ> 0 and 1 ∈ L1(Ω) under the homogeneous Dirichlet boundary condition. The function f is assumed to be continuous and non-decreasing with f (0) = 0. As is well-known, L1 theory of Brezis and Strauss ([2]) is of great use to deal with this problem. Below, we shall see that FVM is a suitable discretization method for this problem in the sense that the discrete version of [2] can be applied. Consequently, we immediately deduce the generation of the nonlinear semigroup, namely, the unique existence of a time global solution to a semidiscrete (in space) FVM for a degenerate parabolic equation of the form ut −∆ f (u) = 0. Then, we readily obtain stability results in L1 and L∞, and order-preserving property for finite volume solutions by the nonlinear semigroup theory. This is totally new approach to study FVM for degenerate elliptic and parabolic problems. As an application, we shall consider a degenerate Ke

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