Numeerisen analyysin ja laskennallisen
tieteen seminaari
10.10.2005 klo
14.15
U356
Mika Juntunen and Rolf Stenberg
A Finite Element
Method For General Boundary Condition
Enforcing perturbed Dirichlet boundary condition i.e. the Robin
boundary condition with small coefficient in the derivative term leads
to a high condition number in the system matrix. Perturbed boundary
condition also plagues the adaptive mesh refinement based on the a
posteriori error estimate since the straight forward formulation of the
problem leads into a posteriori estimate that induces a too dense mesh
on the boundary. A numerical scheme has to take these facts into
account in order to produce an e cient and numerically stable method.
In this note we introduce a method based on the Nitsche method [1] [2]
[3] to circumvent the high condition number of the system matrix in the
case of the perturbed boundary condition. The method is proposed in a
way that it is possible to move continuously between the Neumann and
the Dirichlet boundary conditions. We show that the method is
consistent and prove the a priori error estimate which shows that the
method has the optimal rate of convergence. Under the saturation
assumption we also prove the a posteriori error estimate. The proposed
a posteriori error estimate is efficient since it is also a lower bound
of the error.
References
[1] J.A. Nitsche. Über ein
Variationsprinzip zur Lösung von Dirichlet-Problemen bei
Verwendung von Teilräumen, die keinen Randbedingungen unterworfen
sind. Abhandlungen aus dem Mathematischen Seminar der
Universität Hamburg, 36:9 15, 1970/71.
[2] Rolf Stenberg. Mortaring by a
method of J.A. Nitsche. Computational Mechanics; New Trends and
Applications, S. Idelsohn, E. Oate and E. Dvorkin (Eds.), @CIMNE,
Barcelona, Spain, 1998.
[3] Roland Becker, Peter Hansbo, and Rolf Stenberg. A finite element method for domain
decomposition with non-matching grids. Mathematical Modelling
and Numerical Analysis, 37(2):209 225, 2003.