A Godunov‐Mixed Finite Element Method on Changing Meshes for the Nonlinear Sobolev Equations

Tongjun Sun

doi:10.1155/2012/413718

Abstract

A Godunov‐mixed finite element method on changing meshes is presented to simulate the nonlinear Sobolev equations. The convection term of the nonlinear Sobolev equations is approximated by a Godunov‐type procedure and the diffusion term by an expanded mixed finite element method. The method can simultaneously approximate the scalar unknown and the vector flux effectively, reducing the continuity of the finite element space. Almost optimal error estimates in L2‐norm under very general changes in the mesh can be obtained. Finally, a numerical experiment is given to illustrate the efficiency of the method.

Highlights

We consider the following nonlinear Sobolev equations: ut − ∇ · a x, t, u ∇ut b x, t, u ∇u c x, t, u · ∇u f x, t, u, x ∈ Ω, t ∈ 0, T, u x, t 0, x ∈ ∂Ω, t ∈ 0, T, 1.1 u x, 0 u0 x, x ∈ Ω, where Ω is a bounded subset of Rn n ≤ 3 with smooth boundary ∂Ω, u0 x and f x, t, u are known functions, and the coefficients a x, t, u, b x, t, u, c x, t, u c1 x, t, u, c2 x, t, u T
Advection was approximated by a Godunov-type procedure, and diffusion was approximated by a low-order-mixed finite element method
The object of this paper is to present a Godunov-mixed finite element method on changing meshes for the nonlinear Sobolev equations

Summary

Introduction

We consider the following nonlinear Sobolev equations: ut − ∇ · a x, t, u ∇ut b x, t, u ∇u c x, t, u · ∇u f x, t, u , x ∈ Ω, t ∈ 0, T , u x, t 0, x ∈ ∂Ω, t ∈ 0, T , 1.1 u x, 0 u0 x , x ∈ Ω, where Ω is a bounded subset of Rn n ≤ 3 with smooth boundary ∂Ω, u0 x and f x, t, u are known functions, and the coefficients a x, t, u , b x, t, u , c x, t, u c1 x, t, u , c2 x, t, u T satisfy the following condition:. Methods which combined Godunov-type schemes for advection with mixed finite elements for diffusion were introduced in 14 and had been applied to flow problems in reservoir engineering, contaminant transport, and computational fluid dynamics. Applications of these types of methods to single and two-phase flow in oil reservoirs were discussed in 15, ; application to the Navier-Stokes equations was given in. The convection term c · ∇u of the nonlinear Sobolev equations is approximated by a Godunov-type procedure and the diffusion term by an expanded mixed finite element method This method can simultaneously approximate the scalar unknown and the vector flux effectively, reducing the continuity of the finite element space. Throughout the analysis, the symbol K will denote a generic constant, which is independent of mesh parameters Δt and h and not necessarily the same at different occurrences

Hm div fx 2 m fy

Projections and Lemma

Error Estimates

Numerical Example