Biharmonic wave equations are of importance to various applications including thin plate analyses. In this work, the numerical approximation of their solutions by a C 1 -conforming in space and time finite element approach is proposed and analyzed. Therein, the smoothness properties of solutions to the continuous evolution problem is embodied. High potential of the presented approach for more sophisticated multi-physics and multi-scale systems is expected. Time discretization is based on a combined Galerkin and collocation technique. For space discretization the Bogner-Fox-Schmit element is applied. Optimal order error estimates are proven. The convergence and performance properties are illustrated with numerical experiments.
The quasi-static multiple network poroelastic theory (MPET) model, first introduced in the context of geomechanics, has recently found new applications in medicine. In practice, the parameters in the MPET equations can vary over several orders of magnitude which makes their stable discretization and fast solution a challenging task. Here, a new efficient parameterrobust hybridized discontinuous Galerkin method, which also features fluid mass conservation, is proposed for the MPET model. Its stability analysis which is crucial for the well-posedness of the discrete problem is performed and cost-efficient fast parameter-robust preconditioners are derived. We present a series of numerical computations for a 4-network MPET model of a human brain which support the performance of the new algorithms.Key words and phrases. MPET model, strongly mass-conserving high-order discretizations, parameter-robust LBB stability, norm-equivalent preconditioners, hybrid discontinuous Galerkin methods, hybrid mixed methods.JK, ML and KO acknowledge the funding by the German Science Fund (DFG) -project "Physics-oriented solvers for multicompartmental poromechanics" under grant number 456235063. JS acknowledges the funding by the Austrian Science Fund (FWF) through the research programm "Taming complexity in partial differential systems" (F65) -project "Automated discretization in multiphysics" (P10). PL acknowledges the support by the Academy of Finland (Decision 324611).
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