A wavelet-based stabilization of the mixed finite element method with Lagrange multipliers

We present a new stabilized mixed finite element method for second order elliptic equations in divergence form with Neumann boundary conditions. The approach introduces first the trace of the solution on the boundary as a Lagrange multiplier, which yields a corresponding residual term that is expressed in the Sobolev norm of order 1/2 by means of wavelet bases. The stabilization procedure is then completed with the residuals arising from the constitutive and equilibrium equations. We show that the resulting mixed variational formulation and the associated Galerkin scheme are well posed. In addition, we provide a residual-based reliable and efficient a posteriori error estimate.