Jim Douglas

ALTERNATING-DIRECTION GALERKIN METHODS ON RECTANGLES

Publisher Summary This chapter presents an overview of alternating-direction Galerkin methods on rectangles. Alternating-direction methods in several forms have proved to be very valuable in the approximate solution of partial differential equations problems involving several space variables by finite differences. The methods have been applied to transient problems directly and to stationary problems as iterative procedures. The chapter presents highly efficient procedures for the numerical solution of second-order parabolic and hyperbolic problems in two or more space variables and for the iterative solution of the algebraic equations arising from the Galerkin treatment of elliptic problems. The results presented are limited to rectangular domains. The chapter presents heat equation on a rectangle and extensions to variable coefficients and nonlinear parabolic equations and systems. It describes an iterative procedure for elliptic equations.

Some superconvergence results for an H1 - Galerkin procedure for the heat equation

Abstract : SDouglas,Jim , Jr.;Dupont,Todd ;Wheeler,Mary Fanett ;MRC-TSR-1382DA-31-124-ARO(D)-462Sponsored in part by National Science Foundation.*Heat transfer, *Partial differential equations, Calculus of variations, Convergence, Approximation, Theorems*Galerkin method, Parabolic differential equations, Heat equationThomee and Wahlbin have introduced a Galerkin method for the heat equation in a single space variable based on the (H sup 1)-inner product and have obtained (H sup 2) and (H sup 1) estimates for the error. An (L sup 2) estimate is given here. The main object is to show knot superconvergence phenomena when the subspace is a piecewise-polynomial space. For (C sup 2)-piecewise-polynomials of degree r, the error in the knot values is O(h sup(2r-2)); for the (C sup 1) case, both knot values and knot first x-derivatives are approximated to within O(h sup(2r-2)). (Author)