Cell Discretization of Nonselfadjoint Linear Elliptic Partial Differential Equations

Abstract

The cell discretization (CD) algorithm is generalized to handle nonselfadjoint elliptic partial differential equations. A primal-dual functional is used as the basis for the discretization process, combined with the standard moment collocation for interface constraints which was used for selfadjoint equations. The unknown function is expanded, as usual, as a linear combination of basis functions within each subdomain, or cell, while the moment collocation uses weight functions defined over the interfaces. The transformation of discrete variables which simplifies the interface conditions, coupled with the conjugacy condition on the transformation coefficients, carries over to the nonselfadjoint case and decouples the intracell and interface variables. This makes almost all the computations independent cell-by-cell, rendering it very suitable for parallelizing. Two model convection-diffusion problems are solved with diffusivities down to .0001. The results are presented in the form of surface plots, showing the effect of increasing the number of collocations and cell parameters and the use of margins to contain boundary layers. Comparisons are made among the three conjugate direction iterative procedures used to solve the discrete equations. An effort is made to relate the CD procedure to various classical mathematical relationships.