Abstract
In this paper we consider numerical solution methods for two dimensional Fredholm integral equation of the second kind f ( x , y ) − ∫ − 1 1 ∫ − 1 1 a ( x , y , u , v ) f ( u , v ) d u d v = g ( x , y ) , ( x , y ) ∈ [ − 1 , 1 ] × [ − 1 , 1 ] , where a ( x , y , u , v ) is smooth and g ( x , y ) is in L 2 [ − 1 , 1 ] 2 . We discuss polynomial interpolation methods for four-variable functions and then use the interpolating polynomial to approximate the kernel function a ( x , y , u , v ) . Based on the approximation we deduce fast matrix-vector multiplication algorithms and efficient preconditioners for the above two dimensional integral equations. The residual correction scheme is used to solve the discretization linear system.
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