Geodesic
Anisotropic mesh refinement in polyhedral domains: error estimates with data in L 2 (Ω)
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 4, pp. 1117-1145

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The paper is concerned with the finite element solution of the Poisson equation with homogeneous Dirichlet boundary condition in a three-dimensional domain. Anisotropic, graded meshes from a former paper are reused for dealing with the singular behaviour of the solution in the vicinity of the non-smooth parts of the boundary. The discretization error is analyzed for the piecewise linear approximation in the H1(Ω)- and L2(Ω)-norms by using a new quasi-interpolation operator. This new interpolant is introduced in order to prove the estimates for L2(Ω)-data in the differential equation which is not possible for the standard nodal interpolant. These new estimates allow for the extension of certain error estimates for optimal control problems with elliptic partial differential equations and for a simpler proof of the discrete compactness property for edge elements of any order on this kind of finite element meshes.

DOI : 10.1051/m2an/2013134
Classification : 65N30
Keywords: elliptic boundary value problem, edge and vertex singularities, finite element method, anisotropic mesh grading, optimal control problem, discrete compactness property 
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     author = {Apel, Thomas and Lombardi, Ariel L. and Winkler, Max},
     title = {Anisotropic mesh refinement in polyhedral domains: error estimates with data in $L^2(\Omega )$},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1117--1145},
     publisher = {EDP-Sciences},
     volume = {48},
     number = {4},
     year = {2014},
     doi = {10.1051/m2an/2013134},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2013134/}
}
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Apel, Thomas; Lombardi, Ariel L.; Winkler, Max. Anisotropic mesh refinement in polyhedral domains: error estimates with data in $L^2(\Omega )$. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 4, pp. 1117-1145. doi: 10.1051/m2an/2013134

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