Oscillation-preserving Legendre-Galerkin methods for second kind integral equations with highly oscillatory kernels

Abstract

The original solutions of highly oscillatory integral equations usually have rapid oscillation, which means that conventional numerical approaches used to solve these equations have poor convergence. In order to overcome this difficulty, in this paper, we propose and analyze an oscillation-preserving Legendre-Galerkin method for second kind integral equations with highly oscillatory kernels. Concretely, we first incorporate the standard approximation subspace of Legendre polynomial basis with a finite number of oscillatory functions which capture the oscillation of the exact solutions. Then, we construct an efficient numerical integration scheme, yielding a fully discrete linear system. Making use of best approximation results for some weighted projection operators defined in suitable weighted Sobolev spaces and the compactness operator theory, we establish that the fully discrete approximate equation has a unique solution and the approximate solution reaches an optimal convergence order without the influence of the wave number. In addition, we prove that for sufficiently large wave number, the spectral condition number of the corresponding linear system is uniformly bounded. At last, we use numerical examples to demonstrate the effectiveness of the proposed method.

Approximate Solution of a Class of Highly Oscillatory Integral Equations Using an Exponential Fitting Collocation Method

“…We employ the proposed collocation methods for solving HOVIE (45) and report the errors for several values of N in Table 3. We also plot (in logarithmic scale) the errors embedded in Table 3 for both methods in Figure 3.…”