Summary. Given 2>0 and p~(0, 1), we consider the following problem: findwhere ~r~c~-~2 is a smooth convex domain. We prove optimal H 1 and L ~ error bounds for the standard continuous piecewise linear Galerkin finite element approximation. In addition we analyse a more practical approximation using numerical integration on the nonlinear term. Finally we consider a modified nonlinear SOR algorithm, which is shown to be globally convergent, for solving the algebraic system derived from the more practical approximation.

Summary. This paper considers a fully practical piecewise linear finite element approximation of the Dirichlet problem for a second order self-adjoint elliptic equation, Au=f, in a smooth region f2cN" (n=2 or 3) by the boundary penalty method. Using an unfitted mesh; that is f~h, an approximation of f2 with dist (f2, f2 h) < C h 2, is not in general a union of elements; and assuming ueH+(f2) we show that one can recover the total flux across a segment of the boundary of O with an error of O(hZ). We use these results to study a fully practical piecewise linear finite element approximation of an elliptic equation by the boundary penalty method when the prescribed data on part of the boundary is the total flux.