By Marián Vajtersic
This quantity offers with difficulties of recent powerful algorithms for the numerical answer of the main usually happening elliptic partial differential equations. From the perspective of implementation, realization is paid to algorithms for either classical sequential and parallel computers.
the 1st chapters are dedicated to quickly algorithms for fixing the Poisson and biharmonic equation. within the 3rd bankruptcy, parallel algorithms for version parallel desktops of the SIMD and MIMD kinds are defined. The implementation facets of parallel algorithms for fixing version elliptic boundary price difficulties are defined for structures with matrix, pipeline and multiprocessor parallel laptop architectures. a latest and well known multigrid computational precept which deals a great chance for a parallel cognizance is defined within the subsequent bankruptcy. extra parallel editions established during this notion are offered, wherein equipment and assignments thoughts for hypercube structures are handled in additional aspect. The final bankruptcy provides VLSI designs for fixing precise tridiagonal linear platforms of equations coming up from finite-difference approximations of elliptic difficulties.
For researchers drawn to the advance and alertness of quick algorithms for fixing elliptic partial differential equations utilizing complex computers.
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Extra info for Algorithms for Elliptic Problems: Efficient Sequential and Parallel Solvers
2 k +1 - , 1 22 Chapter I where j i = i2[+ I, = I, 2, ... , "}k-r - L 1'=0,1, ... ) ) -I _ (2p( r) _ q ( r) ) PN+1 N+I-2' N+l' . 26) tliat alle! 27). 2 Problem with periodic boulldary value cOllditiollS Approximation of t he solution of the Poisson cCJuatioll y) = f(x, y) ~u(x, on a rectangle, with periodic boundary conditions U(xo, y) = u(:rN+l, y), u(x,yo) = u(x, YN+l), by central differences leads to a system of linear equations, expressed in matrix form Mpu = v. :31) so that the Fourier transform v = Q -I V compu tes according to form llJap 2 = -N +1 V'JI V21+1 L N+I 2 VjCOS i=1 N+l = N +1 L ,=1 Vi (211") li- ' sin N N +1 1= 1, 2, ...
The eigenvalues of the positive ddinite matrix Rare determined front the explicitly known eigeuvallles 0 < PI :S P2 ... :S Pm of the matrix :B by Jlj = 1 - P~-j+I. The idea of Stiefel's method far a given initial u(O) amI a residuum r(O) is to find such vectors u(k) that the residuum r(k) = v - Ru(k) be expresseel in the form r(k) = Qj(R)r(O), j = 1, 2, .... Q j (:r) is a polynomial of degree j defined by where R o< III :S = (I 112 ... rst kind allel a = 2/(0:ß), b = (0: + ß)/(o: - ß) with 0: and ß being the upper alld lower bounds, respectively, for thc eigenvalues Il.
So), H So = (-1,2,-l), = (-1,21,1). 46), we obtain the Peacernall- Hachford iteration parameters for j == 1, 2, ... , s, where A (0,414) 2 s :S A~ . 47) Chapter 1 30 where r(k) is the residuum vector, (lk anel bk are the respective roefficients. 47), based on a. romputation using Chebyshev polynomials, is ak = Ao +2 Al bk • ao = 1, b-1 k bo = 4/(Ao = Ao +2 AI _ (Ao - AI ) 2 b. 4 k-l, k = 1,2, ... , + At). There exist several variants ofthe Jacobi, Gauss-Seidel allel SOR methoels that improve their properties.
Algorithms for Elliptic Problems: Efficient Sequential and Parallel Solvers by Marián Vajtersic