A discontinuous \(hp\) finite element method for diffusion problems: 1-D analysis (Q1963025)

The mathematical background for a new type of discontinuous Galerkin method is reported, including stability analysis and error estimates, and is illustrated by the numerical analysis of a steady state 1-D diffusion problem. The shape functions are discontinuous at the element boundaries. The main features of the method are: auxiliary variables are not needed; robustness; the method yields elementwise conservative approximations; resulting matrices are positive-definite, well conditioned and convenient for the treatment of transient problems; adaptive error control; acceptable costs.