h, p, k Least Squares Finite Element Processes for 1-D Helmholtz Equation
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The paper presents mathematical details of the finite element processes using the Galerkin method with weak form and least square processes for 1-D Helmholtz equation with Robin boundary condition in h,p,k framework. The concepts of variational consistency (VC) and variational inconsistency (VIC) are discussed. It is shown that Galerkin method with or without weak form is variationally inconsistent (VIC). The VIC of the Galerkin method is due to non-zero wave number ( term) as well as Robin boundary condition. These two aspects of VIC of the integral forms and their consequences are investigated individually as well as jointly. It is shown that the integral forms yield non-symmetric functional B(. ,.) and hence the possibility of complex basis for the coefficient matrix for some choices of h, p and k, which may lead to spurious or oscillatory non-physical numerical solutions. It is demonstrated that the variational consistency of the Galerkin method or galerkin method with weak form cannot be restored through any mathematically justifiable means, hence spuriousness in the computed solution is inevitable for those ranges of physical and computational parameters for which basis of the coefficient matrix becomes complex, though may possibly be minimized using different approaches reported in the literature. Least square processes, on the other hand, are always variationally consistent for any wave number and any choice of computational parameters. Hence, the resulting coefficient matrices are always symmetric, positive definite and have real basis and thus are free of spurious numerical solutions. Periodic theoretical solutions of most model problems, including those used here, are of higher order global differentiability and hence necessitate use of higher order spaces for local approximations. Mathematical details as well as numerical studies are presented to illustrate various features of the proposed approach.