Analysis For A Kind Of Meshless Galerkin Method And The Lower Approximation Of Eigenvalues

This paper consists of two chapters. In the first chapter.it is firstly shown that the discrete form of RKPM is independent of the nodal quadrature weights and that MLSM can be regarded as a by-product of the discrete form of RKPM,if nodal integration is applied to discretize the continuous form of RKPM. Main object of this chapter is to investigate error estimates of RKPM and generic hp-cloud methods. To this end, a special partition of unity function space in which inverse inequalities can be established, is constructed. Based on the special partition of unity function space, a generic k-consistency relation for RKPM and a p-consistency relation for hp-cloud method have been found. We also show that Galerkin approximation solutions based on RKPM and hp-cloud methods converge. As meshless methods, the convergence rates are measured by new control variable ,i.e. radius of influence domains of weights. In the numerical examples, second elliptic boundary problems are solved by using hp-cloud methods and a very fast convergence rate is observed with respect to various norms.In the second chapter, we reveal two interestingly and theoretically important properties of lumped mass finite element approximation eigenvalues.i.e.. lumped mass finite element approximation eigenvalues are always smaller than the exact eigenvalues and that they are increasing functions of the number of total elements. For model eigenvalue problems, we rigorously prove these propreties for uniform mesh. And for general eigenvalue problems, we originally propose an equivalent form of lumped mass finite element scheme and demonstrate that lumped mass finite element eigenvalues are always smaller than standard finite element approximation eigenvalues. Our numerical experiments demonstrate all theoretical results and show that lumped mass finite element approximation eigenvalues are increasing functions of the number of total elements, combining these results with results from standard finite element method, we obtain conclusions for general problem.