Implementation of Karhunen-Loeve expansion using discontinuous Legendre polynomial based Galerkin approach

This work presents a numerical Galerkin scheme based on discontinuous Legendre polynomials (DLPG) to solve the integral eigenvalue problem known as Fredholm integral equation of a second kind which is mainly used for random fields representation by means of Karhunen-Loeve expansion. The main advantages of the proposed method are the simple applicability of constructing the Legendre bases over each local elemental domain without considering any continuity between the elements, in addition to the orthogonality properties of the Legendre polynomials. The latter result in a fast, easy assembly and sparse representation of functions forming the linear system of equations, unlike the conventional Galerkin methods which tend to be computationally demanding. Three covariance functions are examined to demonstrate the feasibility and accuracy of the proposed method. The convergence properties of the approximated eigenvalues and the second moment of the numerically approximated random fields using the DLPG approach are confirmed with h-and p-refinement regarding a one-dimensional example. Furthermore, the applicability and efficiency is shown for two-dimensional domains considering both, rectangular and arbitrary domain shapes.