Additive Schwarz algorithms for solving hp-version finite element systems on triangular meshes
Applied Numerical Mathematics
A parallel spectral element method for dynamic three-dimensional nonlinear elasticity problems
Computers and Structures
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In this paper, the development and implementation of highly parallelizable domain decomposition solvers for adaptive hp finite element methods is discussed. Two-level orthogonalization is used to obtain a reduced system which is preconditioned by a coarse grid operator. The condition number of the preconditioned system, for Poisson problems in two space dimensions, is proved to be bounded by C(1 + Hp/h)2(1 + log p)2 and Cp(1 + Hp/h)2(1 + log p)2 for different choices of coarse grid operators, where H is the subdomain size, p is the maximum spectral order, h is the size of the smallest element in the subdomain, and C is a constant independent of the mesh parameters. The work here extends the work of Bramble et al. [Math Comp., 47 (1986), pp. 103--134] on the h-version and Babuska et al. [SIAM J. Numer. Anal., 29 (1991), pp. 624--661] on the p-version of the finite element method. A preliminary version of this solver was first announced by Oden, Patra, and Feng in [Domain Decomposition Solver for Adaptive hp Finite Elements, VII Conference on Domain Decomposition, State College, PA, October 1993]. Numerical experiments show fast convergence of the solver and good control of the condition number on a variety of discretizations.