Local H − 1 Galerkin and adjoint local H − 1 Galerkin procedures for elliptic equations

— Two essentially dual, finite element methods for approximating the solution of the boundary value p rob le m Lu =• V ( Ö V « ) + b • V u + cu — f on Q, a rectangle•, with u = O on dCl are shown to give optimal order convergence. The local H~ method is based on the inner product («, L* v) and the adjoint method on {Lu, v). Discontinuons spaces can be employed for the approximate solution in the local H' procedure and for the test space in the adjoint method.