We design and analyze a Hybrid High-Order (HHO) method on unfitted meshes to approximate elliptic interface problems by means of a consistent penalty methodà la Nitsche. The curved interface can cut through the mesh cells in a rather general fashion. Robustness with respect to the cuts is achieved by using a cell agglomeration technique, and robustness with respect to the contrast in the diffusion coefficients is achieved by using a different gradient reconstruction on each side of the interface. A key novel feature of the gradient reconstruction is to incorporate a jump term across the interface, thereby releasing the Nitsche penalty parameter from the constraint of being large enough. Error estimates with optimal convergence rates are established. A robust cell agglomeration procedure limiting the agglomerations to the nearest neighbors is devised. Numerical simulations for various interface shapes corroborate the theoretical results.
We study the existence of strong solutions to a 2d fluid-structure system. The fluid is modelled by the incompressible Navier–Stokes equations. The structure represents a steering gear and is described by two parameters corresponding to angles of deformation. Its equations are derived from a virtual work principle. The global domain represents a wind tunnel and imposes mixed boundary conditions to the fluid velocity. Our method reposes on the analysis of the linearized system. Under a compatibility condition on the initial data, we can guarantee local existence in time of strong solutions to the fluid-structure problem.
We design and analyze an arbitrary-order hybridized discontinuous Galerkin method to approximate the unique continuation problem subject to the Helmholtz equation. The method is analyzed using conditional stability estimates for the continuous problem, leading to error estimates in norms over interior subdomains of the computational domain. The convergence order reflects the Hölder continuity of the conditional stability estimates and the approximation properties of the finite element space for sufficiently smooth solutions. Under a certain convexity condition, the constant in the estimates is independent of the frequency. Moreover, certain weighted averages of the error are shown to converge independently of the stability properties of the continuous problem. Numerical examples illustrate the performances of the method with respect to the degree of ill-posedness of the problem, increasing polynomial order, and perturbations in the data.
We design and analyze a hybrid high-order method on unfitted meshes to approximate the Stokes interface problem. The interface can cut through the mesh cells in a very general fashion. A cell-agglomeration procedure prevents the appearance of small cut cells. Our main results are inf-sup stability and a priori error estimates with optimal convergence rates in the energy norm. Numerical simulations corroborate these results.
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