Computer Methods in Applied Mechanics and Engineering - Special edition on the 20th Anniversary
A Finite Element Variational Multiscale Method for the Navier-Stokes Equations
SIAM Journal on Scientific Computing
SIAM Journal on Numerical Analysis
Stabilization of Low-order Mixed Finite Elements for the Stokes Equations
SIAM Journal on Numerical Analysis
Applied Numerical Mathematics
A Generalization of the Local Projection Stabilization for Convection-Diffusion-Reaction Equations
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
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Local projection stabilization (LPS) of finite element methods is a new technique for the numerical solution of transport-dominated problems. The main aim of this paper is a critical discussion and comparison of the one- and two-level approaches to LPS for the linear advection-diffusion-reaction problem. Moreover, the paper contains several other novel contributions to the theory of LPS. In particular, we derive an error estimate showing not only the usual error dependence on the mesh width but also on the polynomial degree of the finite element space. Based on this error estimate, we propose a definition of the stabilization parameter depending on the data of the solved problem. Unlike other papers on LPS methods, we observe that the consistency error may deteriorate the convergence order. Finally, we explain the relation between the LPS method and residual-based stabilization techniques for simplicial finite elements.