A new way of devising numerical methods is introduced whose distinctive feature is the computation of a finite element approximation only in a polyhedral subdomain D of the original, possibly curved-boundary domain. The technique is applied to a discontinuous Galerkin method for the one-dimensional diffusion-reaction problem. Sharp a priori error estimates are obtained which identify conditions, on the subdomain D and the discretization parameters of the discontinuous Galerkin method, under which the method maintains its original optimal convergence properties. The error analysis is new even in the case in which D=Ω. It allows to see that the uniform error at any given interval is bounded by an interpolation error associated to the interval plus a significantly smaller error of a global nature. Numerical results confirming the sharpness of the theoretical results are displayed. Also, preliminary numerical results illustrating the application of the method to two-dimensional second-order elliptic problems are shown.
Bibliographical noteFunding Information:
B. Cockburn was supported in part by the National Science Foundation (Grant DMS-0712955) and by the University of Minnesota Supercomputing Institute.
- Discontinuous Galerkin methods
- Elliptic problems