A Numerical Approach to a Nonlinear and Degenerate Parabolic Problem by Regularization Scheme

Haitao Cao

doi:10.4236/jamp.2014.25012

Abstract

In this work we propose a numerical scheme for a nonlinear and degenerate parabolic problem having application in petroleum reservoir and groundwater aquifer simulation. The degeneracy of the equation includes both locally fast and slow diffusion (i.e. the diffusion coefficients may explode or vanish in some point). The main difficulty is that the true solution is typically lacking in regularity. Our numerical approach includes a regularization step and a standard discretization procedure by means of C0-piecewise linear finite elements in space and backward-differences in time. Within this frame work, we analyze the accuracy of the scheme by using an integral test function and obtain several error estimates in suitable norms.

Highlights

Denoting Ω ⊂ Rn (n ≥ 1) as the domain occupied by the porous medium with Lipschitz boundary and (0, T] (0 < T < +∞) as the time interval
In this work we propose a numerical method to the following nonlinear and degenerate parabolic equation
We point out that the Equation (1.1) is usually obtained after using the Kirchhoff transformation. Such problems can be investigated through a parabolic regularization of function b(u) or by perturbing the boundary and initial data such that the corresponding solutions do not take the degenerate points

Summary

Introduction

Denoting Ω ⊂ Rn (n ≥ 1) as the domain occupied by the porous medium with Lipschitz boundary and (0, T] (0 < T < +∞) as the time interval. We point out that the Equation (1.1) is usually obtained after using the Kirchhoff transformation Such problems can be investigated through a parabolic regularization of function b(u) or by perturbing the boundary and initial data such that the corresponding solutions do not take the degenerate points. In [8] [9], Richards’ equation is analyzed by a numerical approach consisting in a regularization procedure and discretization by means of C0-piecewise linear finite elements (or mixed finite elements) in space and backward-differences in time. The second step is a standard discretization procedure by means of C0-piecewise linear finite elements in space and backward-differences in time Within this frame work, we analyze the accuracy of the scheme by using an integral test function and obtain several error estimates in suitable norms. C will denote a generic positive constant which is independent of ε

Problem Setting and Main Result

Proof of Main Result