Combinations of collocation and finite-element methods for Poisson's equation

In this paper, we provide a framework of combinations of collocation method (CM) with the finite-element method (FEM). The key idea is to link the Galerkin method to the least squares method which is then approximated by integration approximation, and led to the CM. The new important uniformly V⁰_h-elliptic inequality is proved. Interestingly, the integration approximation plays a role only in satisfying the uniformly V⁰_h-elliptic inequality. For the combinations of the finite-element and collocation methods (FEM-CM), the optimal convergence rates can be achieved. The advantage of the CM is to formulate easily linear algebraic equations, where the associated matrices are positive definite but nonsymmetric. We may also solve the algebraic equations of FEM and the collocation equations directly by the least squares method, thus, to greatly improve numerical stability. Numerical experiments are also carried for Poisson's problem to support the analysis. Note that the analysis in this paper is distinct from the existing literature, and it covers a large class of the CM using various admissible functions, such as the radial basis functions, the Sinc functions, etc.