Domain decomposition methods for coupled Stokes-Darcy flows
Changqing Wang · D-Scholarship@Pitt (University of Pittsburgh) · 2017
This thesis studies the numerical methods for coupled Stokes-Darcy problem. It consists of three major parts: First, a non-overlapping domain decomposition method is presented for Stokes-Darcy problem by partitioning the computational domain into multiple subdomains, upon which families of coupled local problems of lower complexity are formulated. The coupling is based on appropriate interface matching conditions. The global problem is reduced to an interface problem by eliminating the interior subdomain variables, which can be solved by an iterative procedure. FETI approach is used for floating Stokes subdomains. The condition number of the resulting algebraic system is analyzed and numerical tests on matching grids verifying the theoretical estimates are provided. Second, a multiscale flux basis algorithm is developed based on the domain decomposition with multiscale mortar mixed finite element method. The algorithm involves precomputing a multiscale flux basis, which consists of the flux (or velocity trace) response from each mortar degree of freedom. It is computed by each subdomain independently before the interface iteration begins. The subdomain solves required at each iteration are substituted by a linear combination of the multiscale basis. This may lead to a significant reduction in computational cost since the number of subdomain solves is fixed, depending only on the number of mortar degrees of freedom associated with a subdomain. Several numerical examples are carried out to demonstrate the efficiency of the multiscale flux basis implementation. Third, a multiscale flux basis implementation is presented for coupled Stokes-Darcy flows with stochastic permeability, with its log represented as a sum of local Karhunen-Lo\`{e}ve expansions. The problem is approximated by stochastic collocation on either a tensor product or a sparse grid, coupled with multiscale mortar mixed finite element method using non-overlapping domain decomposition for the spatial discretization. Two algorithms based on deterministic or stochastic multiscale flux basis are introduced. Some numerical tests are presented to illustrate the performances of these algorithms, with the stochastic multiscale flux basis showing a great advantage in computational cost among all.