Nonlinear elliptic problems with the method of finite volumes

Sanjay Kumar Khattri

doi:10.1155/denm/2006/31797

Abstract

We present a finite volume discretization of the nonlinear elliptic problems. The discretization results in a nonlinear algebraic system of equations. A Newton‐Krylov algorithm is also presented for solving the system of nonlinear algebraic equations. Numerically solving nonlinear partial differential equations consists of discretizing the nonlinear partial differential equation and then solving the formed nonlinear system of equations. We demonstrate the convergence of the discretization scheme and also the convergence of the Newton solver through a variety of practical numerical examples.

Highlights

Nonlinear elliptic equations arise in many applications in many fields, so solving such systems is important
Solving nonlinear partial differential equations consists of discretizing the nonlinear partial differential equation and solving the formed nonlinear system of equations
Solving nonlinear partial differential equations consists of discretizing the partial differential equations and solving the formed nonlinear algebraic system of equations

Summary

Introduction

Nonlinear elliptic equations arise in many applications in many fields, so solving such systems is important. Work has been done on numerically solving nonlinear elliptic partial differential equations (PDEs). We explore the convergence of the discretization method, and the convergence behaviour of the Newton-Krylov method for solving the nonlinear algebraic equations [7, 8]. This paper presents the two-point finite volume discretization (2P-FVM) [9, 10] of the nonlinear problem (1.1) and (1.2) on the rectangular meshes. Several numerical examples are reported for showing the convergence of the finite volume discretization scheme. A Newton-Krylov algorithm is mentioned for solving the system of nonlinear equations formed by the discretization scheme.

Two-point finite volume discretization of the nonlinear elliptic problem

Finite volume for nonlinear elliptic problems

Newton-Krylov algorithm

Numerical examples

Conclusions