This talk presents some new and renovated finite element methods (FEMs) that aim at solving Darcy, elasticity, and poroelasticity.  The novel weak Galerkin (WG) finite element methods use degree k(>=0) polynomials in element interiors and on edges separately for approximating the primal variable (pressure).  The discrete weak gradients of these weak basis functions are established in the Arbogast-Correa spaces, which are improvements of the classical Raviart-Thomas spaces.  These discrete weak gradients are then used to approximate the classical gradient in the variational form.  These new FEMs do not use any nonphysical penalty but are locally conservative and produce continuous normal fluxes.

Furthermore, these new methods have optimal-order convergence in pressure, velocity, normal flux, and div of velocity.  The Bernardi-Raugel elements, originally designed for Stokes flow, can be re-used for elasticity, after incorporation of the techniques of reduced integration.  These methods will be combined to solve poroelasticity.  Numerical examples will be presented to illustrate these ideas.

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