Convergence analysis of a domain decomposition method for separable PDEs

Gongyuan Zhuang, Purdue University

Abstract

We consider a complicated partial differential equation (PDE) problem involving multiple PDE operators defined on a complicated domain. One approach to solve the PDE problem is to split the domain into simple subdomains, each with just one PDE operator. Interface conditions must be introduced along boundaries between subdomains, the usual ones are continuity of the solution and its normal derivative. One can then define iteration methods where each step involves solving the simpler PDE problems on the subdomains. Such methods are called non-overlapping domain decomposition methods using an interface relaxation iteration. In this thesis, we analyze a two-step method called the “Dirichlet/Neumann Averaging” method. In this method, the PDE problems defined on the subdomains are solved with Dirichlet interface conditions at odd iterations, then a relaxation formula is used to smooth the normal derivatives along the interface, then the subproblems are solved with Neumann interface conditions at even iterations, and finally we use another relaxation formula to smooth the Dirichlet values along the interface. After making an initial guess, these actions define one iteration with two steps of the Dirichlet/Neumann Averaging (AVE) method. This AVE method has been analyzed previously by Rice, Vavalis and Yang for the case of two rectangular subdomains and a single Helmholtz operator. There are two parameters in the relaxation formulas and it was shown that the iteration converges for certain conditions on the parameters. In this thesis, we use eigenfunction expansions to find the optimum (or good) parameters for this AVE method, and prove convergence for the larger class of separable PDE operators. We extend the analysis to some variations. It is known experimentally that interface relaxation methods converge for a much broader class of PDE problems than one can analyze. With the help of SciAgents and MATLAB, we Verify our theoretical results with numerical experiments. We also illustrate by examples that the results are likely to be true in much more general cases.

Degree

Ph.D.

Advisors

Rice, Purdue University.

Subject Area

Mathematics

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