ON THE STABILITY OF THE L PROJECTION IN H(Ω)

We prove the stability in H1(Ω) of the L2 projection onto a family of finite element spaces of conforming piecewise linear functions satisfying certain local mesh conditions. We give explicit formulae to check these conditions for a given finite element mesh in any number of spatial dimensions. In particular, stability of the L2 projection in H1(Ω) holds for locally quasiuniform geometrically refined meshes as long as the volume of neighboring elements does not change too drastically.

[1]  L. R. Scott,et al.  Finite element interpolation of nonsmooth functions satisfying boundary conditions , 1990 .

[2]  Olaf Steinbach,et al.  Boundary element preconditioners for a hypersingular integral equation on an interval , 1999, Adv. Comput. Math..

[3]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[4]  O. Steinbach.  Adaptive nite element – boundary element solution of boundary value problems , 1999 .

[5]  Olaf Steinbach,et al.  On a hybrid boundary element method , 2000, Numerische Mathematik.

[6]  Jinchao Xu,et al.  Some Estimates for a Weighted L 2 Projection , 1991 .

[7]  P. Clément.  Approximation by finite element functions using local regularization , 1975 .

[8]  Panayot S. Vassilevski,et al.  Computational scales of Sobolev norms with application to preconditioning , 2000, Math. Comput..

[9]  Olaf Steinbach,et al.  A new a posteriori error estimator in adaptive direct boundary element methods: the Dirichlet problem , 2000 .

[10]  Abdellatif Agouzal,et al.  Une méthode d'éléments finis hybrides en décomposition de domaines , 1995 .

[11]  V. Thomée,et al.  The stability in _{} and ¹_{} of the ₂-projection onto finite element function spaces , 1987 .

[12]  Olaf Steinbach,et al.  The construction of some efficient preconditioners in the boundary element method , 1998, Adv. Comput. Math..

[13]  L. Wahlbin.  Superconvergence in Galerkin Finite Element Methods , 1995 .