Each $H^{1/2}$-stable projection yields convergence and quasi-optimality of adaptive FEM with inhomogeneous Dirichlet data in $R^d$

Aurada, M.; Feischl, M.; Kemetmüller, J.; Page, M.; Praetorius, D.

doi:10.1051/m2an/2013069

Each

H^{1 / 2}

-stable projection yields convergence and quasi-optimality of adaptive FEM with inhomogeneous Dirichlet data in

R^{d}

Aurada, M. ; Feischl, M. ; Kemetmüller, J. ; Page, M. ; Praetorius, D.

ESAIM: Mathematical Modelling and Numerical Analysis , Direct and inverse modeling of the cardiovascular and respiratory systems. Numéro spécial, Tome 47 (2013) no. 4, pp. 1207-1235

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Résumé

We consider the solution of second order elliptic PDEs in R^d with inhomogeneous Dirichlet data by means of an h-adaptive FEM with fixed polynomial order p ∈ N. As model example serves the Poisson equation with mixed Dirichlet-Neumann boundary conditions, where the inhomogeneous Dirichlet data are discretized by use of an H^1 / 2-stable projection, for instance, the L²-projection for p = 1 or the Scott-Zhang projection for general p ≥ 1. For error estimation, we use a residual error estimator which includes the Dirichlet data oscillations. We prove that each H^1 / 2-stable projection yields convergence of the adaptive algorithm even with quasi-optimal convergence rate. Numerical experiments with the Scott-Zhang projection conclude the work.

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DOI : 10.1051/m2an/2013069

Classification : 65N30, 65N50
Keywords: adaptive finite element method, convergence analysis, quasi-optimality, inhomogeneous Dirichlet data

Aurada, M.; Feischl, M.; Kemetmüller, J.; Page, M.; Praetorius, D. Each $H^{1/2}$-stable projection yields convergence and quasi-optimality of adaptive FEM with inhomogeneous Dirichlet data in $R^d$. ESAIM: Mathematical Modelling and Numerical Analysis , Direct and inverse modeling of the cardiovascular and respiratory systems. Numéro spécial, Tome 47 (2013) no. 4, pp. 1207-1235. doi: 10.1051/m2an/2013069

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     title = {Each $H^{1/2}$-stable projection yields convergence and quasi-optimality of adaptive {FEM} with inhomogeneous {Dirichlet} data in $R^d$},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1207--1235},
     year = {2013},
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