Parallel Domain Decomposition Solver for Adaptive HP Finite Element Methods

• Published in: SIAM journal on numerical analysis Vol. 34; no. 6; pp. 2090 - 2118
• Main Authors: Oden, J. T., Patra, Abani, Feng, Yusheng
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.12.1997
• Subjects: Applied mathematics; Approximation; Boundary value problems; Decomposition; Degrees of freedom; Degrees of polynomials; Exact sciences and technology; Finite element method; Mathematical analysis; Mathematics; Methods; Methods of scientific computing (including symbolic computation, algebraic computation); Numerical analysis; Numerical analysis. Scientific computation; Orthogonality; Partial differential equations; Partial differential equations, boundary value problems; Polynomials; Preconditioning; Sciences and techniques of general use; Shape functions; Vertices; Finite element method; Grid pattern; Decomposition method; Parallel computation; Numerical method; Domain decomposition; Adaptive method; Preconditioning; Implementation
• ISSN: 0036-1429; 1095-7170
• DOI: 10.1137/S0036142994278887

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 + log Hp/h)2(1 + log p)2and Cp(1 + log Hp/h)2(1 + log p)2for 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.