## A Black-Box Iterative Solver Based on a Two-Level Schwarz by Brezina M., Vanek P.

By Brezina M., Vanek P.

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Ruge, J. : Algebraic multigrid (AMG) for sparse matrix equations. In: Sparsity and its applications (Evans, D. ), pp. 257–284. Cambridge: Cambridge University Press, Cambridge 1985. : Robust iterative solvers on unstructured meshes. D. thesis University of Colorado at Denver 1997. [4] Chan, T. : Domain decomposition and multigrid algorithms for elliptic problems on unstructured meshes. , UCLA, 1993. CAM Report. [5] Chan, T. : Domain decomposition for unstructured mesh problems. In: Proceedings of the seventh international symposium on domain decomposition methods for partial differential equations (Keys, D.

The complex problem above can be easily reformulated as a real one with two real pressures p R , p I and their gradients uR , uI . The continuous problem has been discretized using uniform Q1 finite elements. As the components of the gradient of a Q1function are not Q1-functions themselves, the discretization of an integral \ω ∇p − u 2 d (44) creates an undesirable “artificial viscosity” resulting in damping of the numerical solution. To avoid this, (44) has been discretized using piecewise constant quadrature formula applied elementwise.

Vanˇek and the boundary integral enforces the radiation boundary condition ∂p = ikp ∂r on ∂ . This is why all the integrals in (43) vanish for the minimizer of F . The complex problem above can be easily reformulated as a real one with two real pressures p R , p I and their gradients uR , uI . The continuous problem has been discretized using uniform Q1 finite elements. As the components of the gradient of a Q1function are not Q1-functions themselves, the discretization of an integral \ω ∇p − u 2 d (44) creates an undesirable “artificial viscosity” resulting in damping of the numerical solution.