Error Bounds of a Fully Discrete Projection Method for Symm’s Integral Equation

Abstract

The approximation properties of a fully discrete projection method for Symm’s integral equation with a infinite smooth boundary have been investigated. For the method, error bounds have been found in the metric of Sobolev’s spaces. The method turns out to be more accurate compared to the fully discrete collocation method known before.

On accuracy of solving Symm's equation by a fully discrete projection method

“…In the paper [13] it was established that with the selection of discretization parameters by the rules…”

“…;m;ı;" u n;m;ı;" k Ä 2z 5 . /n C1 [13] and in Theorem 6.1. It is evident that the main orders by ı and " in (4.10) and (6.1) are the same.…”

“…Here, in the metric space H 0 it is found an error bound of FDPM for the case of perturbed input data and known smoothness of the solution. Later in the scale of spaces H for all 0 Ä < the accuracy of the method (3.6) was established in [13]. In the next section we make more precise some results from the above mentioned papers [8,13], which we will need further.…”

“…In distinction from [14] the present paper is dedicated to minimizing an accuracy bound of the fully discrete projection method for solving Symm's equation in the case where smoothness of solution is unknown. Besides, the main differences our paper with [13] are the following. Now we consider the other kind of kernel error representation (see Remark 6.3) and as the stopping rule we use the balancing principle.…”

“…The investigation of the second method continued in [13], where error bounds were found in the metric of Sobolev's spaces. Further in a posteriori case the error bounds of fully discretized collocation method in the scale of Sobolev spaces were established in [14].…”