### A direct boundary integral method for a mobility problem.

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Many discrepancy principles are known for choosing the parameter α in the regularized operator equation $(T*T+\alpha I){x}_{\alpha}^{\delta}=T*{y}^{\delta}$, $|y-{y}^{\delta}|\le \delta $, in order to approximate the minimal norm least-squares solution of the operator equation Tx = y. We consider a class of discrepancy principles for choosing the regularization parameter when T*T and $T*{y}^{\delta}$ are approximated by Aₙ and $z{\u2099}^{\delta}$ respectively with Aₙ not necessarily self-adjoint. This procedure generalizes the work of Engl and Neubauer (1985), and particular cases of the results are applicable...

The aim of this paper is to give a convergence proof of a numerical method for the Dirichlet problem on doubly connected plane regions using the method of reflection across the exterior boundary curve (which is analytic) combined with integral equations extended over the interior boundary curve (which may be irregular with infinitely many angular points).

Mathematics Subject Classification: 44A40, 45B05The paper presents an abstract linear second kind Fredholm integral equation with degenerated kernel defined by means of the Bittner operational calculus. Fredholm alternative for mutually conjugated integral equations is also shown here. Some examples of solutions of the considered integral equation in various operational calculus models are also given.

We propose an adaptive finite element method for the solution of a linear Fredholm integral equation of the first kind. We derive a posteriori error estimates in the functional to be minimized and in the regularized solution to this functional, and formulate corresponding adaptive algorithms. To do this we specify nonlinear results obtained earlier for the case of a linear bounded operator. Numerical experiments justify the efficiency of our a posteriori estimates applied both to the computationally...