A stabilizer free weak Galerkin method for the Biharmonic Equation on Polytopal Meshes

A new stabilizer free weak Galerkin (WG) method is introduced and analyzed for the biharmonic equation. Stabilizing/penalty terms are often necessary in the finite element formulations with discontinuous approximations to ensure the stability of the methods. Removal of stabilizers will simplify finite element formulations and reduce programming complexity. This stabilizer free WG method has an ultra simple formulation and can work on general partitions with polygons/polyhedra. Optimal order error estimates in a discrete $H^2$ for $k\ge 2$ and in $L^2$ norm for $k>2$ are established for the corresponding weak Galerkin finite element solutions. Numerical results are provided to confirm the theories.Comment: arXiv admin note: text overlap with arXiv:1309.5560, arXiv:1510.06001 by other author

C1C^1-conforming variational discretization of the biharmonic wave equation

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