L2L^2-error estimates for finite element approximations of boundary fluxes

We prove quasi-optimal a priori error estimates for finite element approximations of boundary normal fluxes in the $L^2$-norm. Our results are valid for a variety of different schemes for weakly enforcing Dirichlet boundary conditions including Nitsche's method, and Lagrange multiplier methods. The proof is based on an error representation formula that is derived by using a discrete dual problem with $L^2$-Dirichlet boundary data and combines a weighted discrete stability estimate for the dual problem with anisotropic interpolation estimates in the boundary zone.Comment: 16 pages, 3 figure

An a posteriori error analysis for the equations of stationary incompressible magnetohydrodynamics

Magnetohydrodynamics (MHD) is a continuum level model for conducting fluids subject to external magnetic fields, e.g. plasmas and liquid metals. The efficient and robust solution of the MHD system poses many challenges due to it's nonlinear, non self-adjoint, and highly coupled nature. In this paper, we develop a robust and accurate a posteriori error estimate for the numerical solution of the MHD equations based on the exact penalty method. The error estimate also isolates particular contributions of error in a quantity of interest (QoI) to inform discretization choices to arrive at accurate solutions. The tools required for these estimates involve duality arguments and computable residuals