By Marián Vajtersic
This quantity bargains with difficulties of contemporary powerful algorithms for the numerical answer of the main often taking place elliptic partial differential equations. From the perspective 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 forms are defined. The implementation facets of parallel algorithms for fixing version elliptic boundary worth difficulties are defined for structures with matrix, pipeline and multiprocessor parallel computing device architectures. a contemporary and renowned multigrid computational precept which deals a great chance for a parallel consciousness is defined within the subsequent bankruptcy. extra parallel editions established during this concept are offered, wherein equipment and assignments ideas for hypercube structures are handled in additional element. The final bankruptcy offers VLSI designs for fixing distinctive tridiagonal linear structures of equations bobbing up from finite-difference approximations of elliptic difficulties.
For researchers drawn to the improvement and alertness of quickly algorithms for fixing elliptic partial differential equations utilizing complicated computers.
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Additional resources for Algorithms for Elliptic Problems: Efficient Sequential and Parallel Solvers
Example text
Ors whose components correspond to the points of the grid Rh( 1) alld Rh( 2), respectively. The matrices E and F correspond to the five-point approximation on the subdomains and the matrix S represents the functional elependence between the components of the solution. (l) . ( 1) , , u(2):::: Ud2 u2(2))] r . (1). Thell u( 1) and u( 2) are computed by Fast methods for solving the Poisson equation u(l) = E- 1v(l) - E- 1 u(2) = F-1 V (2) - m "1(2), F- [Sfl ",(1). 68), the formulae ui(l) = zi(1) - Zi(1)u1(2), ui(2) = zi(2) - Zi(2)u r (1), i = 1, 2, ...
58) by y = v + e. (5) Solving of Mu = y. 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).
30). Table 3 Algorithm Complexity Matrix decomposition with Fourier trallsform SN'21 og N Cyclic reductioll ; N 2 10g N BUlleman 6 N 2 10g N Table :3 gives tllP evaluations of compntational cOl1lplexity for all three considered algorithms. 1 Introcludion Having construeted the eliscretizatioll grid witl! te step h = l/(N + 1) on a rectangular domain, after approximating the Diriehlet problem for separablc elliptic partial equations by final elifferenccs at the grid points, we obtain linear systems of equations of the form Cu= w, (LH) where C is a positive definite matrix of elimension ;V2.