# Chapter 5 Numerical Solution of Elliptic Problems in One Dependent Variable

- Publisher
- Elsevier Science & Technology
- Identifiers
- DOI: 10.1016/s0376-7361(08)70238-3
- Disciplines

## Abstract

Publisher Summary This chapter presents the numerical solution of elliptic problems in one dependent variable. A sparse matrix is one in which most of the elements are zero. Finite-difference methods yield sparse matrices that allow special techniques to be used for their solution. An upper triangular matrix is one in which all the elements below and to the left of the main diagonal are zero; a lower triangular matrix is one in which all the elements above and to the right of the main diagonal are zero. A diagonal matrix has zero elements on all but the main diagonal. The chapter discusses an algorithm for the solution of band equations by the method of factorization. The types of iteration for the solution of elliptic problems––(1) relaxation, (2) alternating-direction, and (3) strongly implicit procedure––are discussed. All three types are used throughout the industry, and any given simulator may use one or more of these types. All iterative methods involve making some initial guess for the dependent variable or variables.

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