Abstract
In this paper, we will discuss on the Combined Legendre spectral-Finite element methods (CLSFEM) for the two-dimensional Fredholm integral equations with smooth kernel on the Banach spaces and the corresponding eigenvalue problem. In these methods, the approximated finite dimensional space is the cartesian product of spline space and Legendre polynomial space. The problem is approximated by the CLSFEM using orthogonal projection, which projects from the Banach space into the finite dimensional space. The convergence analysis for both Fredholm integral equations and the corresponding eigenvalue problem will be discussed in both L 2 and norms. The numerical results will be shown to validate the theoretical estimate.
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