A Parallel High-Order Solver for Linear Elasticity Problems Using A Weak GalerkinFinite Element Method on Unstructured Quadrilateral Meshes Open Access
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Weak Galerkin Finite Element Method (WGFEM)  is a high-order discontinuous method for unstructured grids. Recently, The WGFEM has been successfully applied to solve linear elasticity problems. In this work, we integrate a domain decomposition method with a WGFEM solver for parallel solutions of linear elasticity equations on unstructured quadrilateral grids. In particular, we employ a Balancing Domain Decomposition by Constraints (BDDC)  to effectively reduce the computational cost of the coarse problem for the interface unknowns. The WG-BDDC method shares some similarity with the well-known FETI-DP. However, the standard continuous Galerkin Finite Element Method solves unknowns on nodal points. These nodal points on subdomain interfaces may connect to multiple elements in the FETI-DP method after domain decomposition. The WG-BDDCmethod does not need to involve solutions on the nodal points. Therefore, the communication between two adjacent subdomains only requires to collect information over their common faces.A unique feature of the WGFEM method is the use of weak gradient operator to differentiated basis functions. For 2D problems, an integration by parts was used to change surface integrals to line integrals. WGFEM employs two kinds of basis functions. The first group is on the edges of each element and the second group is inside each element. Two groups of basis function can be chosen independently. In order to have continuous solution in each element given that two different spaces are used, a stabilizer is introduced at element interfaces. In the BDDC method, the unknowns are grouped to interior, dual and primal spaces for each subdomain by the technique of Schur complement. Unknowns in the primal spaces will constitute a new global problem to solve. The size of this new global problem is significantly smaller than that of the global problem involving all unknowns on subdomain interfaces. First, we tested the orders of accuracy of WG-BDDC method on structured grids. Subsequently, by using our university cluster, the WG-BDDC is robust for solving linear elasticity problems on parallel computers. Excellent scalability performance was obtained for the BDDC method on uniform grids by testing over 144 CPUs. Moreover, the WG-BDDCmethod is successfully extended to fully unstructured grids of all quadrilateral elements to solve beam deformation problems. The results on unstructured grids are comparable qualitatively to that on structured grids.