Weighted error estimates of the continuous interior penalty method for singularly perturbed problems

Abstract

Abstract.In this paper we analyze local properties of the Continuous Interior Penalty (CIP) Method for a model convection-dominated singularly perturbed convection-diffusion problem. We show weighted a priori error estimates, where the weight function exponentially decays outside the subdomain of interest. This result shows that locally, the CIP method is comparable to the Streamline Diffusion (SD) or the Discontinuous Galerkin (DG) methods.

A finite element time relaxation method

“…These methods are designed so as to have optimal convergence for smooth solutions [9,8,2,4], and to contain perturbations in an O (h) region from layers. This approach is appealing for large eddy simulation since it indicates the possibility of (i) optimal convergence for smooth solutions [4]; (ii) containment of pollution caused by high frequency content due to the nonlinear coupling [5]; (iii) optimal dissipation rates in the turbulent zone [3,10]. Similar quasi-optimal convergence proofs were obtained recently for a finite element realization of the time-relaxation method [7] in the simple case of a linear transport equation.…”

Implicit-explicit Runge–Kutta schemes and finite elements with symmetric stabilization for advection-diffusion equations

“…All these methods lead to similar L 2 -norm error estimates for smooth solutions, resulting in the loss of half a power of h in the advection-dominated regime (compared to a full power in the unstabilized case). For solutions with sharp layers, it has been proven for SUPG [23], DG [21], and CIP [12] that quasi-optimal convergence is retained away from layers, hence prohibiting the global spreading of spurious oscillations.…”

Robust Numerical Methods for Singularly Perturbed Differential Equations

“…An alternative way of avoiding the assumption of quasi-uniformity is to replace the L 2 projection i h by the standard nodal interpolation u I . Local error estimates similar to those stated for the streamline diffusion method in Theorem 3.41 have been established in [BGL07]. ♣ Remark 3.88.…”