The aim of this talk is to present highly stable collocation based numerical methods for Volterra Integral Equations (VIEs). As it is well known, a collocation method is based on the idea of approximating the exact solution of a given integral equation with a suitable function belonging to a chosen finite dimensional space, usually a piecewise algebraic polynomial, which satisfies the integral ...

We consider a boundary element collocation method for the heat equation. As trial functions we use the tensor products of continuous piecewise linear splines with collocation at the nodal points. Convergence and stability is proved in the case where the spatial domain is a disc. Moreover, practical implementation is discussed in some detail. Numerical experiments confirm our results. Introducti...

We present a collocatIon method based on piecewise cubic polynomials I In C for solving various Integral and differential equatIons. Two new methods for handling curved boundaries are discussed based on collocation and discrete least-squares. Finally, a summary of the properties of the collocation method Is given which IndIcates the simplicity, flexlbl1fty, efficiency, and generality of the met...

A research code has been written to solve an elliptic system of coupled nonlinear partial differential equations of conservation form on a rectangularly shaped three-dimensional domain. The code uses the method of collocation of Gauss points with tricubic Hermite piecewise continuous polynomial basis functions. The system of equations is solved by iteration. The system of nonlinear equations is...

We consider solving integral equations on a piecewise smooth surface S in R 3 with a smooth kernel function, using piecewise polynomial collocation interpolation of the surface. The theoretical analysis includes the eeects of the numerical integration of the collocation coeecients and the use of the approximating surface. The resulting order of convergence is higher than had previously been exp...

We discuss an a-posteriori error estimate for the numerical solution of boundary value problems for nonlinear systems of ordinary differential equations with a singularity of the first kind. The estimate for the global error of an approximation obtained by collocation with piecewise polynomial functions is based on the defect correction principle. We prove that for collocation methods which are...

We propose and study numerically the implicit approximation in time of Navier–Stokes equations by a Galerkin–collocation method combined with inf–sup stable finite element methods space. The conceptual basis approach is establishment direct connection between Galerkin classical collocation methods, perspective achieving accuracy former reduced computational costs terms less complex algebraic sy...

The commonly used graded piecewise polynomial collocation method for weakly singular Volterra integral equations may cause serious round-off error problems due to its use of extremely nonuniform partitions and the sensitivity of such time-dependent equations to round-off errors. The singularity preserving (nonpolynomial) collocation method is known to have only local convergence. To overcome th...

A nonlinear Dirichlet boundary value problem is approximated by an orthogonal spline collocation scheme using piecewise Hermite bicubic functions. Existence, local uniqueness, and error analysis of the collocation solution and convergence of Newton’s method are studied. The mesh independence principle for the collocation problem is proved and used to develop an efficient multilevel solution met...