By Marián Vajtersic
This quantity offers with difficulties of contemporary powerful algorithms for the numerical resolution of the main usually happening elliptic partial differential equations. From the viewpoint of implementation, realization is paid to algorithms for either classical sequential and parallel desktops.
the 1st chapters are dedicated to quick 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 elements of parallel algorithms for fixing version elliptic boundary price difficulties are defined for platforms with matrix, pipeline and multiprocessor parallel machine architectures. a contemporary and well known multigrid computational precept which bargains an outstanding chance for a parallel cognizance is defined within the subsequent bankruptcy. extra parallel editions dependent during this suggestion are offered, wherein tools and assignments suggestions for hypercube platforms are handled in additional element. The final bankruptcy provides VLSI designs for fixing targeted tridiagonal linear structures of equations coming up from finite-difference approximations of elliptic difficulties.
For researchers attracted to the advance and alertness of quick algorithms for fixing elliptic partial differential equations utilizing complicated desktops.
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Extra resources for Algorithms for Elliptic Problems: Efficient Sequential and Parallel Solvers
S1Ich right-hancl-side vectors are produccc! l')rj(Y). In lh(' complllalion 01' step 1 wc obtain the veclor Qz first. anel tl]p1l the vec( or Qv, = Vi = kiQz. hc marching; methocl As we have alrcacly mcntionecl in tile illtroductioll 10 Ihis chapter, the fastest marching algorithm for solving tlle Poisson cCjllatioll was developed by Lorcl1z . It assumes that Cl. rectangular domain, wilh Cl. grid having N X N cliscretization ste])s, is decom posed into L s u bdomains. 1 = 0, 1, ... , L - 1.
72). Another method for solving the Dirichlet problem on a domain composed of rectangles was proposed by Kuznetsov . The solution on a given domain is constructed by all iterative method, llsing decomposition into suitable sllbspaces. The method of conjugate gradients is used with advantage to compute the solutions of the subsystems. 5], where E is the prescribed accuracy of the solution. g. integral equations). 5 An algorithm für an octagonaJ clomaill As an example of applying the cmbcdding lllethod we describe the fast algorithm for solving the POiSSOll equatioll Oll an octagonal domain, suggested by Temperton .
3) Compute the vectors QVj, Uj i = 1,2, ... = QUj, i , N. = 1, 2, ... , N. Step 3 is not perfonned by direct multiplication of a vector by a matrix, ])+1 hut is decomposed, for positive integers q1, q2,··· ,qp+l, qi ~ 2, into the followillg phases: (a) for s = p + 1, (h) if s ]J, . • . , = ]J + 1 then if s f; p + 1 thell 1 L qi = N -1, i=] Chapter I 14 ( c) for TI = 2, :3, .... (j" - 1 s = 2: qi· where js i=1 Provided that N is apower of 2, step 1 may be pel'fol'lned by applying the FFT algorithIll in O(N 2 10g N) operations.