Galerkin Collocation Research Articles

A Galerkin collocation method for some integral equations of the first kind

Here we present a certain modified collocation method which is a fully discretized numerical method for the solution of Fredholm integral equations of the first kind with logarithmic kernel as principal part. The scheme combines high accuracy from Galerkin's method with the high speed of collocation methods. The corresponding asymptotic error analysis shows optimal order of convergence in the sense of finite element approximation. The whole method is an improved boundary integral method for a wide class of plane boundary value problems involving finite element approximations on the boundary curve. The numerical experiments reveal both, high speed and high accuracy.

A Collocation–Galerkin Method for the Two Point Boundary Value Problem Using Continuous Piecewise Polynomial Spaces

A collocation–Galerkin method for the two point boundary value problem based on continuous piecewise polynomial spaces is defined where the collocation points are the roots of a Jacobi polynomial. An existence and uniqueness theorem is derived by means of a related variational procedure that is defined using a semi-discrete innerproduct. Optimal rates of convergence are established and $O(h^{2r} )$ order of superconvergence at the nodes is obtained. Additional approximations to the solution and its derivatives that obtain $O(h^{2r} )$ order of superconvergence at any point are described.

Some Collocation-Galerkin Methods for Two-Point Boundary Value Problems

A family of discontinuous and simply continuous collocation–Galerkin methods corresponding to the $H^{ - 1} $ methods of Rachford and Wheeler is defined. $L_p $-estimates optimal in local step size and norm on solution are obtained, and local quadratures giving accuracy of higher order are developed.