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Num3: Numerical Mathematics and Scientific Computing Location: A703 Session Chair: Stefan Frei | |
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Stability and Error Analysis of Unfitted Isoparametric Scott--Vogelius Elements for Stokes Flows 1: Department of Mathematics, University of Pittsburgh, PA, USA; 2: Department of Mathematics, University of Houston, TX, USA; 3: Institute of Mathematics, Friedrich-Schiller-Universität Jena, Germany In this talk, we analyse a higher-order unfitted finite element method for the incompressible Stokes equations, yielding a pointwise divergence-free solution in the entire domain. The approach is based on the isoparametric Scott--Vogelius velocity-pressure pair on a background mesh, together with a stabilized mixed Nitsche/Lagrange multiplier formulation for imposing Dirichlet boundary conditions. We use a higher-order Lagrange multiplier space to ensure stability and to mitigate the loss of pressure robustness typically associated with the weak enforcement of Dirichlet boundary conditions. The key results are a new inf-sup stability result for the isoparametric Scott--Vogelius pair on unfitted meshes and a combined inf-sup stability result for the bilinear forms associated with the pressure and the Lagrange multiplier. We show stability and convergence properties of the method, including geometry approximation errors introduced by the isoparametric approximation. This demonstrates optimal-order velocity convergence of the velocity in the $H^1$- and $L^2$-norms. Furthermore, we establish optimal $H^1$-convergence and nearly optimal $L^2$-convergence of a post-processed pressure. Numerical examples illustrate the theoretical findings. | |
