Equal Higher Order Analysis of an Unfitted Discontinuous Galerkin Method for Stokes Flow Systems

Abstract

In this work, we analyze an unfitted discontinuous Galerkin discretization for the numerical solution of the Stokes system based on equal higher-order discontinuous velocities and pressures. This approach combines the best from both worlds, firstly the advantages of a piece-wise discontinuous high–order accurate approximation and secondly the advantages of an unfitted to the true geometry grid around possibly complex objects and/or geometrical deformations. Utilizing a fictitious domain framework, the physical domain of interest is embedded in an unfitted background mesh and the geometrically unfitted discretization is built upon symmetric interior penalty discontinuous Galerkin formulation. To enhance stability we enrich the discrete variational formulation with a pressure stabilization term. Moreover, the present contribution adopts high order ghost penalty strategies to address the ill conditioning of the system matrix caused by small truncated elements with respect to the unfitted boundary. Motivated by continuous unfitted FEM (Burman and Hansbo in ESAIM Math Model Numer Anal 48(3):859–874, 2014; Massing et al. in J Sci Comput 61:604–628, 2014; Massing et al. in Numer Math 128:73–101, 2014) along with other unfitted mesh surveys grounded on discontinuous spaces (Becker et al. in Comput Methods Appl Mech Eng 198(41–44):3352–3360, 2009; Gürkan and Massing in Comput Methods Appl Mech Eng 348:466–499, 2019; Gürkan et al. in SIAM J Sci Comput 42(5):A2620–A2654, 2020; Massing in A cut discontinuous Galerkin method for coupled bulk-surface problems, Chapter in UCLWorkshop volume on "Geometrically Unfitted Finite Element Methods", Lecture Notes in Computational Science and Engineering, Springer, Cham, pp 259–279, 2017), we use proper velocity and pressure ghost penalties defined on faces of cut cells to establish a robust high-order method, in spite of the cell agglomeration technique usually applied on dG methods. The current presentation should prove valuable in engineering applications where special emphasis is placed on the optimal effective approximation attaining much smaller relative errors in coarser meshes. Inf-sup stability, the optimal order of convergence, and the condition number sensitivity with respect to cut configuration are investigated. Numerical examples verify the theoretical results.