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Parallel solution of the generalized Dirichlet-Neumann map for elliptic PDEs on regular polygon domains

Papadopoulou Eleni, Saridakis Ioannis, Sifalakis Anastasios

Πλήρης Εγγραφή

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Περίληψη

Recently, in [2,3], a new and novel unified approach was introduced for analyzing linear and integrablenonlinear PDEs in two dimensions. Central issue to this approach is a generalized Dirichlet-Neumannmap, characterized through the solution of the so-called global relation, namely an equation, valid forall values of a complex parameter k, coupling specified known and unknown values of the solution andits derivatives on the boundary.For a large class of boundary value problems, the global relation can be solved analytically, and hencethe generalized Dirichlet-Neumann map can be constructed in closed form. However, for general boundaryvalue problems, the global relation must be solved numerically. For this, in [4], a well conditionedand fast convergent collocation-type numerical method was developed and studied for the numerical solutionof the Generalized Dirichlet-Neumann map associated to the generic model problem of Laplace’sequation on an arbitrary convex polygon domain. For the case of regular polygon domains, with thesame type of boundary conditions on all sides, we have (cf. [5]) rigorously studied the properties of theassociated collocation coefficient matrix revealing its Block Circulant structure. And as the block circulantproperty is strongly connected with the Discrete Fourier Transforms (cf. [1]), the produced linearsystem can be solved efficiently using FFTs (cf. [6]). The development of a parallel algorithm for thiscomputational task is the main problem we are addressing in the work herein. The parallel algorithmwe present is realized, through MPI programming, on two parallel systems: (a) on a shared-distributedmemory computer with 8 processors and (b) on a cluster of 4 nodes with 2 processors each. The clusteruses a local ethernet interconnection for its nodes of either 100Mbps or 1 Gbps.Our implementation is further studied through extensive numerical experimentation accompanied withcomputation/communication and speedup measurements (see for example Figure 1 below). Throughthis study we are able to draw conclusions for the performance of our implementation as well as toevaluate and compare the different parallel architectures used.

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