Book Chapter10.1016/B978-0-12-056760-7.50009-8
3 – Elliptic equations
William F. Ames
- 01 Jan 1977
pp 92-164
79
TL;DR: In this paper, the authors consider the problem of finding the solution of an elliptic partial differential equation with auxiliary conditions, which select among all possible solutions, a uniquely determined function, provided that the data is properly posed.
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Abstract: Publisher Summary
Equilibrium problems in two-dimensional, and higher, continua give rise to elliptic partial differential equations. An alternative argument employs the maximum (minimum) modulus theorem. When a partial differential equation has accompanying auxiliary conditions, which select among all possible solutions, a uniquely determined function, the data is called properly posed, provided that the solution depends continuously on this data. Methods of solution for general computational problems fall into two categories—the direct and iterative procedures. Direct methods, of which the solution of a tridiagonal system is typical, give the exact answer in a finite number of steps, if there were no round-off error. The algorithm for such a procedure is complicated and non-repetitive. Many direct methods for linear systems are available. Iterative methods consist of repeated application of a simple algorithm. They yield the answer as a limit of a sequence, even without consideration of round-off errors.
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References
Iterative methods for solving partial difference equations of elliptic type
TL;DR: In this paper, the determinant of the matrix A = (ai,j) does not vanish and if A * = (a*j) is symmetric, where a*1=ai,iai,j/ai,i (i, j= 1, 2, N *, N), then A * is positive definite.