Higher-Order Approximate Solutions of Neumann Problems by Isoparametric Finite Element Method with Relevant Lumping Operator

Abstract

Jn 1966, Friedriehs and Keller [1 j proposed a generalized finite difference scheme for the Neumann problem. Today, their method can be regarded as a finite element method using the piecewise linear chapeau bases. From the viewpoint of computations, however, their method is not convenient enough, except the case of a polygonal domain, since it requires to perform all the integrations which appear in obtaining Ritz coefficients and right-hand sides of determinate linear equations, in an exact domain and on an exact boundary. The purpose of this paper is to consider a higher-order approximate procedure for the Neumann problem in an approximate domain, using isoparametric finite elements. This forces us to change slightly the data functions to gtiarantee the solvability of the determinate linear system. We shall stud some kinds of errors, which are caused by the selection of the basis functions (approximation errors), by the change in domain, by the approximation of data functions and b the change of data functions for the solvability. Errors caused by the selection of die basis functions are, of course, inevitable. Tt is desirable tha t errors due to other reasons mentioned above do not destroy the order of accuracy which the basis functions could achieve. In this sense, it appears to be reasonable to expect that, when the boundary P is curved, the optimal order of accuracy can be achie ed by the use of isoparametric technique. Also, we may apply the lumping technique to the body force term/, which may simplify