Adaptive HP-refinement methods for singularly-perturbed elliptic and parabolic systems.

摘要

Standard Galerkin finite element formulations when applied to singularly perturbed problems, lack the necessary stability and produce solutions featuring non-physical oscillations. We introduce stable high-order finite element methods for convection-diffusion and reaction-diffusion problems. The methods are based on special quadrature rules to achieve stability. The quadrature rules are designed to integrate products of exponential and polynomial functions to high order. In this regard they bear some relationship to exponentially-weighted Petrov-Galerkin methods; however, polynomials are used for both finite element trial and test spaces.; We have development adaptive finite element software that addresses two-dimensional systems of parabolic partial differential equations with capabilities for automatic h-, p- and/or r-refinement. Problems are solved on unstructured triangular meshes. Key components of an adaptive software such as local refinement strategies, projection procedures to transfer computational solutions between finite element meshes, basis functions selection and a posteriori error estimation have been investigated.; Finally, we present a model for the oxidation of a cracked silicon carbide (SiC) matrix that is exposed to a hot gaseous mixture of oxygen ({dollar}Osb2{dollar}) and water vapor ({dollar}Hsb2{dollar}O). The oxidation proceeds into the composite at a rate controlled by the solid-state diffusion of {dollar}Osb2{dollar} and/or {dollar}Hsb2O{dollar} in {dollar}SiOsb2{dollar} to flow and fill the crack. This may reduce damage to the composite. The model consisting of a nonlinear partial differential system, is solved by the adaptive software. Numerical results indicate that adaptive software is an effective and efficient tool to simulate and solve oxidation problems.