Equal higher order analysis of an unfitted discontinuous Galerkin method for Stokes flow systems
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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. A fully stabilized frame is required for equal order finite elements, both –proper for higher-order– pressure Poisson stabilization in the bulk of the domain, as well as boundary zone velocity and pressure ghost penalty terms. The present contribution 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 stabilization parameters dependency are investigated. Our analysis of the stability properties of the proposed scheme reveals that a delicate scaling of the stabilization parameters is required for the equal higher-order case. This is also supported by numerical evidence from test experiments. Additionally, a geometrically robust estimate for the condition number of the stiffness matrix is provided. Numerical examples illustrate the implementation of the method and verify the theoretical findings.
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