An efficient multi projection methods for systems of Fredholm integral equations with mixed weakly singular kernels: A superconvergence approach

In this article, we develop the multi-Galerkin and iterated multi-Galerkin methods to solve systems of second-kind linear Fredholm integral equations (FIEs) with smooth and mixed weakly singular kernels. First, we develop the mathematical formulation of the multi-Galerkin and iterated multi-Galerkin methods using piecewise polynomial approximations to solve such systems and obtain superconvergence results. These methods transform the linear system of FIEs into corresponding matrix equations. We derive error estimates and obtain the convergence analysis. We prove that the convergence rates for the multi-Galerkin method are $O\left(h^{3r}\right)$ for smooth kernels and $O(h^{1+r-\alpha }\log h)$ for mixed weakly singular kernels, where r denote the degree of the piecewise polynomials, h is the norm of partitions and $\alpha =\underset{i,j}{\max }{\alpha }_{ij}$. Moreover, we establish that the iterated multi-Galerkin method achieves improved convergence rates of $O\left(h^{4r}\right)$ for smooth kernels and $O\left(h^{r+2(1-\alpha )}{(\log h)}^2\right)$ for mixed weakly singular kernels. Hence, the results show that the iterated multi-Galerkin method improves the multi-Galerkin method. Finally, the theoretical results are validated through the numerical examples.