Local projection stabilization for advection-diffusion-reaction problems: one-level vs. two-level approach
The authors present a local projection stabilization (LPS) of a finite element method for the linear advection-diffusion-reaction problem \[ Lu:=-\varepsilon \Delta u+{\mathbf b}\cdot\nabla u+\sigma u=f\quad\text{in }\Omega;\qquad u=0\quad \text{on } \partial \Omega, \] where the advection field \({\mathbf b}\) is a finite element solution of an incompressible flow problem. The LPS technique for the numerical solution of transport-dominated probems preserves the stability of methods with residual-based stabilization but has a symmetric form of the stabilization term. The paper contains several novel contributions to the theory of LPS. The authors 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 and propose a definition of the stabilization parameter depending on the data of the solved problem. Finally the authors observe that the consistency error may deteriorate the convergence order and explain the relation between the LPS method and residual-based stabilization techniques for simplicial finite elements.
scientific article; zbMATH DE number 5611785
scientific article