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Additive Schwarz methods for the Crouzeix-Raviart mortar finite element for elliptic problems with discontinuous coefficients. (English) Zbl 1087.65117

The authors are interested in the numerical solution of second order elliptic problems with discontinuous coefficients on a partition \(\bar\Omega=\cup_{j=1}^{p}\bar\Omega_j\) where \(\Omega_j\) is a polygonal subdomain. Homogeneous Dirichlet conditions on the boundary of \(\bar\Omega\) are used and each coefficient \(\rho_j\) (on \(\Omega_j\)) is a positive constant (this problem was considered in many books and papers; even asymptotically optimal algorithms on quasiuniform triangulations were found). But the authors apply the lowest order Crouzeix-Raviart methods for the discretization in each block; the overall discretizations (on nonmatching grids) are modifications of the known mortar procedure.
Their main goal consists in constructing preconditioners on the base of the given partition and the additive Schwarz method for the arising grid systems. The central result relates to a preconditioner \(T^{-1}\) for which the estimate \(c_0\frac{h}{H}a_h(u,u)\leq a_h(Tu,u)\leq c_1a_h(u,u)\) is obtained with positive constants \(c_k\), independent of the coefficients \(\rho_j\) and of the mesh sizes \(h=\min_{j}h_j\) and \(H=\max _{j}H_j\) where \(h_j\) and \(H_j\) correspond to the mesh size of the quasiuniform triangulation and to the diameter of the block \(\bar\Omega_j\). Numerical examples are presented for the case of 16 blocks with each block triangulated into 72 or 50 triangles.

MSC:

65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
35R05 PDEs with low regular coefficients and/or low regular data
65N15 Error bounds for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
65F35 Numerical computation of matrix norms, conditioning, scaling
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