On the $H^{ - 1} $-Galerkin Method for Second-Order Linear Two-Point Boundary Value Problems

Abstract

We discuss several aspects of the $H^{ - 1} $-Galerkin method for the approximate solution of two-point boundary value problems. First we present a formulation of the method for second-order linear equations subject to general linear separated boundary conditions. it is shown that corresponding to such boundary conditions there are adjoint boundary conditions which must be satisfied by elements of the test space. Optimal error estimates for the new $H^{ - 1} $-Galerkin method are derived, a by-product of the analysis being optimal error estimates for the $H^1$-Galerkin method for solving the two-point boundary value problem in question. When B-spline bases are chosen for both the trial (approximation) and test spaces, the linear algebraic systems arising in the $H^{ - 1} $-Galerkin method have a special almost block diagonal structure, which is independent of the form of the boundary conditions. Results of numerical experiments are presented which demonstrate the efficacy of a new package for solving such systems, as well as the predicted orders of convergence of the $H^{ - 1} $-Galerkin procedure. The new package, ROWCOL, is compared with the package SOLVEBLOK [2], and it is shown to be considerably faster than, and as accurate as, SOLVEBLOK for the solution of the linear systems in question. The results derived in this paper can be extended to parabolic equations in one space variable. Also ROWCOL can be used to solve the linear systems arising in the collocation$ - H^{ - 1} $ Galerkin method for two-point boundary value problems and parabolic equations.