Mass preserving finite element implementations of the level set method
Applied Numerical Mathematics - Numerical methods for viscosity solutions and applications
Minimal Stabilization for Discontinuous Galerkin Finite Element Methods for Hyperbolic Problems
Journal of Scientific Computing
Weakly imposed Dirichlet boundary conditions for the Brinkman model of porous media flow
Applied Numerical Mathematics
SIAM Journal on Numerical Analysis
New robust nonconforming finite elements of higher order
Applied Numerical Mathematics
Journal of Scientific Computing
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We discuss stabilized Galerkin approximations in a new framework, widening the scope from the usual dichotomy of the discontinuous Galerkin method on the one hand and Petrov--Galerkin methods such as the SUPG method on the other. The idea is to use interior penalty terms as a means of stabilizing the finite element method using conforming or nonconforming approximation, thus circumventing the need of a Petrov--Galerkin-type choice of spaces. This is made possible by adding a higher-order penalty term giving L2-control of the jumps in the gradients between adjacent elements. We consider convection-diffusion-reaction problems using piecewise linear approximations and prove optimal order a priori error estimates for two different finite element spaces, the standard H1-conforming space of piecewise linears and the nonconforming space of piecewise linear elements where the nodes are situated at the midpoint of the element sides (the Crouzeix--Raviart element). Moreover, we show how the formulation extends to discontinuous Galerkin interior penalty methods in a natural way by domain decomposition using Nitsche's method.