Tensor Product Generalized ADI Methods for Separable Elliptic Problems
Wayne R. Dyksen · SIAM Journal on Numerical Analysis · 1987
We consider solving separable, second order, linear elliptic partial differential equations. If an elliptic problem is separable, then, for certain discretizations, the matrices involved in the corresponding discrete problem can be expressed in terms of tensor products of lower order matrices. In the most general case, the discrete problem can be written in the form $(A_1 \otimes B_2 + B_1 \otimes A_2 )C = F$. We present a new Tensor Product Generalized Alternating Direction Implicit (TPGADI) iterative method for solving such discrete problems. We prove convergence and establish computational efficiency. The TPGADI method is applied to the Hermite bicubic collocation equations. We conclude that the TPGADI method is an effective tool for solving the discrete elliptic problems arising from a large class of elliptic problems.