A posteriori Error Estimates of Galerkin-approxirnate Solutions to the Third Boundary Value Problem for Elliptic Differential Equations

In the present paper we shall show that in the third boundary value problem for elliptic partial differential equations one can get some formulas giving a posteriori error estimates to the approximate solutions obtained by Galerkin's method. Trefftz has proposed in [112] an approximation method which can be used also for getting error estimates to the approximate solutions obtained by Galerkin's method. His method is based on the use of trial functions satisfying the given differential equations. For details of his method, see H3], Ki, D-2]. In practical applications, however, his method is not so convenient, because in general it is not easy to find functions satisfying the conditions requested for the trial functions. On the other hand, in F4T] and Q9], Bramble, Payne and Weinberger gave integral inequalities which can be used for getting error estimates by the use of arbitrary functions satisfying only some smoothness conditions. However the error estimation based on their integral inequalities are not valid, say, when the coefficients of the given differential equation are not continuous. Our method also is based on the use of some trial functions. They