Algorithms for Elliptic Problems: Efficient Sequential and by Marián Vajtersic

By Marián Vajtersic

This quantity bargains with difficulties of recent powerful algorithms for the numerical resolution of the main often taking place elliptic partial differential equations. From the viewpoint of implementation, awareness is paid to algorithms for either classical sequential and parallel computers.
the 1st chapters are dedicated to speedy algorithms for fixing the Poisson and biharmonic equation. within the 3rd bankruptcy, parallel algorithms for version parallel computers of the SIMD and MIMD varieties are defined. The implementation features of parallel algorithms for fixing version elliptic boundary price difficulties are defined for platforms with matrix, pipeline and multiprocessor parallel computing device architectures. a latest and well known multigrid computational precept which bargains a great chance for a parallel recognition is defined within the subsequent bankruptcy. extra parallel variations established during this concept are awarded, wherein equipment and assignments techniques for hypercube structures are taken care of in additional aspect. The final bankruptcy provides VLSI designs for fixing specific tridiagonal linear structures of equations bobbing up from finite-difference approximations of elliptic difficulties.
For researchers drawn to the advance and alertness of quickly algorithms for fixing elliptic partial differential equations utilizing complex computers.

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4 Relaxation multigrid method At present, the so-called multigrid methods, such as [16, 17,40], seem to be advantageous for solving the Poisson equation. One of the algorithms of the lllultigrid methods for solving the Poisson equation on a rectangle is the relaxation algorithm proposed in [3J. In this algorithm, the conditions for its acceleration are based on an analysis of the cOllvergence of the processing over individual grids. The desired solution Oll the finest grid is then obtained by the extrapolation of solutions frolll several grids.

Ion 39 The vector u is the desired solution on Q hand satisfws the boundary value conditlons Oll iJQ n Rh. 2. 3 Method of fictive unknowns Another method for the llumerically solving elliptic boundary value problems based on the embeddillg of a non-rectangular domain into a rectangle is the so-called method of fictive unknowns [1, 20]. The idea of the method is to pxtend the original domain Q to a rectangular domain P and to defillP con veniently both tl](' corresponding elliptic operator in discretized form alld the right-hand-side function at the grid points of the domaill P - Q (the so-ealled fietive components).

Let the difference relations for the unknowll values of the vector Ul at the points of the original domaill be represented by the matrix Lu with the corresponding right-hand-siele vector Wl. 60) is solveel for the right-hand siele v = (WI +LI2W2,L22W2)T, where W2 is a conveniently chosen vector. ion where q(k) = Lu(k) - V, t(k) defined recurrently by = ak+l (I - p(k+l) p(O) = M-1v, a1 = LM- 1 q(k), k bk+IA)p(k) = 1. 60) as follows [23J = p(-l) = 0, U(-l) z(k) u(O) = p(O) = M- 1 v, = M-IRp(k), = (pik), (Rp(k) h) + (u~k) - W2, (Rp(k)h), 1nk = lk + (pik), (Rz(k))d + (ui k ) - W2, (Rz(k) h), if k = 0, lk k = 0, 1, 2, ....

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