On the balancing principle for some problems of Numerical Analysis

We discuss a choice of weight in penalization methods. The motivation for the use of penalization in computational mathematics is to improve the conditioning of the numerical solution. One example of such improvement is a regularization, where a penalization substitutes an ill-posed problem for a well-posed one. In modern numerical methods for PDEs a penalization is used, for example, to enforce a continuity of an approximate solution on non-matching grids. A choice of penalty weight should provide a balance between error components related with convergence and stability, which are usually unknown. In this paper we propose and analyze a simple adaptive strategy for the choice of penalty weight which does not rely on a priori estimates of above mentioned components. It is shown that under natural assumptions the accuracy provided by our adaptive strategy is worse only by a constant factor than one could achieve in the case of known stability and convergence rates. Finally, we successfully apply our strategy for self-regularization of Volterra-type severely ill-posed problems, such as the sideways heat equation, and for the choice of a weight in interior penalty discontinuous approximation on non-matching grids. Numerical experiments on a series of model problems support theoretical results.