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 boundaryzone ]  ] 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 .
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 21,75,76] along with other unfitted mesh surveys grounded on discontinuous spaces 10,44-46,74], 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 .
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high--order
high-order
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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 boundaryzone
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
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21,75,76
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along with other unfitted mesh surveys grounded on discontinuous spaces
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10,44-46,74
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, we use proper
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penalty terms. The present contribution
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
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stabilization parameters dependency
condition number sensitivity with respect to cut configuration
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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
Numerical examples
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findings
results