Abstract
A Galerkin method is presented for calculating the general task weak solution of self-adjoint differential equations with regular singular points, such as the ideal MHD equation for zero-frequency displacements about a finite beta, cylindrical plasma equilibrium (the Newcomb equation). In this case such solutions could be used for constructing the eigenfunctions of resistive instabilities by the method of matched asymptotic expansions. A Galerkin method using finite elements, singular if necessary, is used to calculate the finite energy part of the solution, with the infinite energy part acting as a forcing term. The asymptotic behaviour near the singular point is accurately estimated by postprocessing the Galerkin solution using a generalized Green's function method. The effectiveness of the method is demonstrated on some simple test cases.
Published Version
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