A MONOTONE MULTIDIMENSIONAL UPWIND FINITE ELEMENT METHOD FOR ADVECTION-DIFFUSION PROBLEMS

Abstract

We are interested in developing a multidimensional convective scheme that is capable of dealing with erroneous oscillations near jumps. The strategy is based on the Petrov-Galerkin formulation, to which the underlying idea of the M matrix is added. The nature of the exponentially weighted upwind method is best illuminated by its matrix structure. We interpret the enhanced stability as being due to the attainable irreducible diagonal dominance. The accessible monotonicity condition enables us to construct a monotone stiffness matrix a priori, thereby laying the foundation for arriving at the monotonicity-preserving property. In order to show the merit of the proposed upwinding technique in resolving spurious oscillations generated by unresolved internal and boundary layers, we considered two classes of convection-diffusion problems. As seen from the computed results, we can attain an accurate finite-element solution for a problem free of boundary layer and can capture a high-gradient solution in the sharp layer.

Notes

Address correspondence to Dr. Tony W. H. Sheu, Department of Naval Architecture and Ocean Engineering, National Taiwan University, 73 Chou-Shan Road, Taipei, Taiwan, Republic of China.