A Parallel Preconditioned Bi-Conjugate Gradient Algorithm for Two-Dimensional Elliptic and Parabolic Equations Using Hermite Collocation

A fast and parallelizable method to solve sparse matrix equations that arise from the Hermite collocation discretization of elliptic and parabolic partial di erential equations (PDEs) can be obtained using a bi-conjugate gradient algorithm. A Red-Black GaussSeidel matrix preconditioner provides a structure which makes the algorithm amenable to parallel processing. Increased e ciency is achieved through optimally locating the \collocation points". Results concerning eigenvalues and the e cacy of the method when applied to model problems can be demonstrated. 1 Hermite Cubic Polynomials and Collocation Discretization of PDEs Let u(x; y) be a function de ned on the unit square S = [0; 1] [0; 1]. Partition @S by using m+1 equally spaced nodes fxqg m q=0 and fyrg m r=0 in the xand y-directions, respectively. S is then partitioned into m square elements [xq 1; xq] [yr 1; yr], each of dimension h h, where h = 1 m and where q; r = 1; 2; : : : ; m. Consider the Hermite polynomials, de ned for in the interval [ 2 ; 1 2 ]: fj(x) = 8>< >: 1 2 (1 + 2 ) 2 (1 ); xj 1 x = xj 1 2 + h xj 1 2 (1 2 ) 2 (1 + ); xj x = xj+ 1 2 + h xj+1 0; otherwise gj(x) = 8>< >: h 8 (1 + 2 ) 2 (2 1); xj 1 x = xj 1 2 + h xj h 8 (1 2 ) 2 (2 + 1); xj x = xj+ 1 2 + h xj+1 0; otherwise (1) Then the bi-cubic polynomial interpolating uqr = u(xq; yr), uqr = @u @x (xq; yr), uqr = @u @y (xq; yr), u xy qr = @u @x@y (xq; yr); q; r = 0; 1; 2; : : : ; m; is