It is known for elliptic problems with smooth coefficients
that the solution is smooth in the interior of the domain;
low regularity is only possible near the boundary.
The $hp$-version of the FEM allows us to exploit this
property if we use meshes where the element size grows
porportionally to the element's distance to the boundary
and the approximation order is suitably linked to the
element size. In this way most degrees of freedom are
concentrated near the boundary.
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In this talk, we will discuss convergence and complexity
issues of the boundary concentrated FEM. We will show
that it is comparable to the classical boundary element
method (BEM) in that it leads to the same convergence rate
(error versus degrees of freedom). Additionally, it
generalizes the classical FEM since it does not require
explicit knowledge of the fundamental solution so that
it is also applicable to problems with (smooth) variable
coefficients.