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New and Renovated Finite Elements for Elasticity

It is well known that the continuous Galerkin finite elements suffer Poisson locking when applied to elasticity. In this talk, we first examine the suspicious behaviors of the classical Lagrangian elements in solving linear elasticity problems.  A good remedy is to enrich the Lagrangian elements by edge/face-based bubble functions.  This was motivated by the Bernardi-Raugel elements that were originally designed for Stokes flow.  Then we move on to the novel weak Galerkin finite elements, which use vector-valued polynomial shape functions defined separately in element interiors and on edges/faces.  The discrete weak gradients and divergences of these shape functions are reconstructed via integration by parts in matrix or scalar spaces that have desired approximation properties.  Numerical results along with brief analysis will be presented to demonstrate the accuracy and efficiency of these renovated and novel finite elements.  This talk is based on a series joint work with several collaborators.