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BDDC algorithms for the PDEs with HDG discretizations
Zhang, Jinjin
Zhang, Jinjin
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Abstract
The balancing domain decomposition by constraints methods (BDDC) is one of the most popular nonoverlapping domain decomposition methods. In this dissertation, the BDDC preconditioned conjugate gradient (CG) methods are applied to solve the symmetric saddle point problem such as Stokes problem and Brinkman problem with Hybridizable Discontinuous Galerkin (HDG) discretizations. Edge/face average constraints are enforced across the subdomain interface to ensure that the BDDC preconditioned CG iterations stay in a special subspace, where the preconditioned operator is positive definite. The condition number estimate is given for different choices of stabilization parameters in Stokes problem. Deluxe scaling and adaptively chosen primal constraints are introduced in the Brinkman problem to ensure the condition number is uniformly bounded.The BDDC preconditioned GMRES method is developed for solving the linear system derived from the HDG discretization of Advection-diffusion equations and the Oseen equation of incompressible flow. These problems are either nonsymmetric positive definite or can be reduced to a nonsymmetric positive definite problem in a special subspace. Additional edge constraints are introduced to improve the efficiency. The convergence rates depend on the viscosity. For large viscosity, if the subdomain size is small enough, the number of iterations is independent of the number of subdomains and depends only slightly on the subdomain problem size. The convergence deteriorates when the viscosity decreases.
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Date
2023-05-31
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University of Kansas
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Keywords
Mathematics, Advection diffusion equation, BDDC algorithms, Brinkman equation, HDG discretizations, Oseen equation, Stokes equation