A parallel deflated Krylov solver for finite element problems. |
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| We are using a Finite Element Method [Sepran] to solve various CFD problems, originating from industrial and biological applications. For many applications we are moving from two dimensional problems to three dimensional models. For this problems the memory and CPU time requirements are very high. Therefore we need fast parallel iterative methods in order to compute the solution in reasonable time. For the parallelization of the Finite Element simulation, the grid is decomposed in a number of subgrids. Every element is only part of one subdomain. The construction of the FEM matrix is done per subdomain. Note that these subdomain-matrices are singular for subdomains, which are not connected to a Dirichlet or Robins boundary. The right-hand-side vector is also formed in this way. In order to obtain the global right-hand-side vector we sum up the local right-hand-side vectors in interface points. We use a preconditioned Krylov solver for the linear system. All the vectors used in the solver are stored as global vectors. Preconditioned Krylov solvers consists of four building blocks: vector update, inner product, matrix vector product and preconditioner vector product. For the parallel implementation we use Fortran and MPI. The parallelization of the vector update is straightforward. For the inner product we first form the local inner products keeping into account that the interface values are used more than once. Thereafter a global inner product is formed. The matrix vector product is first done per block. Then the results on the interfaces are sent to the neighboring blocks and summed up. Finally for the preconditioner we use a block variant. We have to keep in mind that some of the subdomain-matrices are singular. The resulting method works fine but is not scalable. When the number of subdomains increases the number of iterations increases. In order to make the method scalable we add a deflation technique. We present some theoretical results and illustrate the method with numerical experiments. |
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algweb@cerfacs.fr Last Update: Apr 17, 2003 |