Weighting the edge stabilization (Q2845607)

The purpose of this article is to propose a nonlinear modification of a standard linear stabilization technique (edge stabilization) that alleviates problems which are encountered with linear stabilization methods. The Gibbs phenomenon, high-order dissipation and convergence to entropy-violating solutions are problems which arise typically.NEWLINENEWLINEIn this paper, the authors focus their attention on the edge stabilization technique as a prototype of linear stabilization which is relatively easy to implement with \(H^1\)-conforming finite elements. Through numerical examples it is shown that edge stabilization can be modified to overcome the usual difficulties, as mentioned above. The main modification proposed consists of introducing a nonlinear mechanism which weakens the edge stabilization in the regions where the discrete solution exhibits large gradients. The near optimal convergence rate is provided and tests on the linear transport equation in one and two space dimensions are done. The latter shows that the weighted edge stabilization performs as required when combined with a nonlinear viscosity method. For polynomial orders larger than or equal to two they observe that the weighted edge stabilization increases the convergence order of the nonlinear method in regions where the solution is smooth.NEWLINENEWLINEAs such, when combining the weighted edge stabilization with a nonlinear viscosity method the convergence order of the nonlinear viscosity method is improved without sacrificing its weakened maximum principle property and its ability to properly converge to entropy solutions.