Numerical solution of nonlinear elliptic partial differential equations by a generalized conjugate gradient method

Journal Article · Sat Dec 31 23:00:00 EST 1977 · Computing; (United States)

DOI:https://doi.org/10.1007/BF02252030· OSTI ID:5786645

Concus, P ^[1]; Golub, G H; O'Leary, D P

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1. Lawrence Berkeley Lab., CA

A generalized conjugate gradient method for solving sparse positive-definite systems of linear equations arising from the discretization of elliptic partial-differential boundary-value problems has been studied previously. Here, extensions to the nonlinear case are considered. The original discretized operator is split into the sum of two operators, one of which corresponds to a more easily solvable system of equations, and the associated iteration based on this splitting is accelerated by (nonlinear) conjugate gradients. The behavior of the method is illustrated for the minimal surface equation with splittings corresponding to nonlinear SSOR, to approximate factorization of the Jacobian matrix, and to elliptic operators suitable for use with fast direct methods. The results of numerical experiments are given as well for a mildly nonlinear example, for which, in the corresponding linear case, the finite termination property of the conjugate gradient algorithm is crucial. 4 tables.

• OSTI ID: 5786645
• Journal Information: Computing; (United States), Journal Name: Computing; (United States) Vol. 19; ISSN CMPTA
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
990200* -- Mathematics & Computers
ALGORITHMS
BOUNDARY-VALUE PROBLEMS
COMPARATIVE EVALUATIONS
COST
DATA
DATA FORMS
DIFFERENTIAL EQUATIONS
EQUATIONS
EXPERIMENTAL DATA
INFORMATION
ITERATIVE METHODS
MATHEMATICAL LOGIC
MATHEMATICAL OPERATORS
NONLINEAR PROBLEMS
NUMERICAL DATA
NUMERICAL SOLUTION
SERIES EXPANSION
SURFACES
TABLES