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Interior penalty preconditioners for mixed finite element approximations of elliptic problems. (English) Zbl 0857.65117
A second-order elliptic problem \[ -\nabla\cdot k\nabla p=f\quad\text{in}\quad \Omega,\quad p= 0\quad\text{on}\quad \partial\Omega \] is reformulated as a mixed problem by introducing \({\mathbf u}=-k\nabla p\): \[ k^{-1}{\mathbf u}+\nabla p=0,\quad \nabla\cdot{\mathbf u}=f. \] This mixed system is to be solved numerically by aid of finite elements of local support and this rises the need for preconditioners for a matrix \(L=BA^{-1}B^*\), where \(B\) is a discrete analogue of \(\nabla\) and \(A\) relates to the matrix \(k^{-1}\). This operator is not local, but by employing an interior penalty method by D. N. Arnold [SIAM J. Numer. Anal. 19, 742-760 (1982; Zbl 0482.65060)] a local operator which is spectrally equivalent to \(L\), may be designed. This may be used for constructing preconditioners.
The method is employed to an application of overlapping domain decomposition methods.

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65F35 Numerical computation of matrix norms, conditioning, scaling
35J25 Boundary value problems for second-order elliptic equations
65F10 Iterative numerical methods for linear systems
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
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