On finite element domain imbedding methods
SIAM Journal on Numerical Analysis
Remarks on mixed finite element methods for problems with rough coefficients
Mathematics of Computation
Finite Element Method for Elliptic Problems
Finite Element Method for Elliptic Problems
The Approximation and Computation of a Basis of the Trace Space H1/2
Journal of Scientific Computing
Benchmark results for testing adaptive finite element eigenvalue procedures
Applied Numerical Mathematics
Hi-index | 0.00 |
We consider a parametric family of boundary value problems for a diffusion equation with a diffusion coefficient equal to a small constant in a subdomain. Such problems are not uniformly well-posed when the constant gets small. However. in a series of papers, Bakhvalov and Knyazev have suggested a natural splitting of the problem into two well-posed problems. Using this idea, we prove a uniform finite element error estimate for our model problem in the standard parameter-independent Sobolev norm. We also study uniform regularity of the transmission problem, needed for approximation. A traditional finite element method with only one additional assumption, namely, that the boundary of the subdomain with the small coefficient does not cut any finite element, is considered.One interpretation of our main theorem is in terms of regularization. Our FEM problem can be viewed as resulting from a Lavrentiev regularization and a Ritz-Galerkin approximation of a symmetric ill-posed problem. Our error estimate can then be used to find an optimal regularization parameter together with the optimal dimension of the approximation subspace.