A high-resolution Petrov-Galerkin method for the convection-diffusion-reaction problem. Part II-A multidimensional extension

A multidimensional extension of the HRPG method (doi:10.1016/j.cma.2009.10.009) using the lowest order block finite elements is presented. First, we design a nondimensional element number that quantifies the characteristic layers which are found only in higher dimensions. This is done by matching the width of the characteristic layers to the width of the parabolic layers found for a fictitious 1D reaction-diffusion problem. The nondimensional element number is then defined using this fictitious reaction coefficient, the diffusion coefficient and an appropriate element size. Next, we introduce anisotropic element length vectors li and the stabilization parameters αi, βi are calculated along these li. Except for the modification to include the new dimensionless number that quantifies the characteristic layers, the definitions of αi, βi are a direct extension of their counterparts in 1D. Using αi, βi and li, objective characteristic tensors associated with the HRPG method are defined. The numerical artifacts across the characteristic layers are manifested as the Gibbs phenomenon. Hence, we treat them just like the artifacts formed across the parabolic layers in the reaction-dominant case. Several 2D examples are presented that support the design objective-stabilization with high-resolution.