TENSOR PRODUCT GENERALIZED ALTERNATING DIRECTION IMPLICIT METHODS FOR SOLVING SEPARABLE SECOND ORDER LINEAR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

WAYNE ROBERT DYKSEN, Purdue University

Abstract

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)(CRTIMES)B(,2) + B(,1)(CRTIMES)A(,2))C = F. We present a new Tensor Product Generalized Alternating Direction Implicit (TPGADI) iterative method for solving such discrete problems. We establish convergence as well as computational efficiency for specific applications in two and three dimensions. In two dimensions, we solve separable elliptic problems on rectangular domains using two types of discretization methods, the Method of Lines and the collocation method. We treat a specific instance of the Method of Lines with Hermite cubic basis functions in the x direction and finite differences in the y direction. A specific instance of the collocation method is presented which uses Hermite bicubic basis functions. In three dimensions, we consider two types of domains, rectangular and cylindrical (nonrectangular in x and y). On rectangular domains, we discretize the partial differential equation by a method which we call the Method of Planes. We treat a specific instance of the Method of Planes with Hermite bicubic basis functions in the xy direction and finite differences in the z direction. On cylindrical domains, we construct the discrete elliptic problem using standard partial difference operators. For each of the four discretization methods mentioned above, we develop the tensor product formulation of the discrete problem and derive a TPGADI method to solve it. We establish convergence and consider the computational complexity and the performance of a specific computer implementation. For each case, we demonstrate that the TPGADI method makes efficient use of computer time and memory. We conclude that the TPGADI method is an effective tool for solving the discrete elliptic problems arising from a large class of elliptic problems on two and three dimensional domains.

Degree

Ph.D.

Subject Area

Mathematics

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