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We introduce and analyze a class of Galerkin-collocation discretization schemes in time for the wave equation. Its conceptual basis is the establishment of a direct connection between the Galerkin method for the time discretization and the classical collocation methods, with the perspective of achieving the accuracy of the former with reduced computational costs provided by the latter in terms of less complex linear algebraic systems. Continuously differentiable in time discrete solutions are obtained by the application of a special quadrature rule involving derivatives. Optimal order error estimates are proved for fully discrete approximations based on the Galerkin-collocation approach. Further, the concept of Galerkin-collocation approximation is extended to twice continuously differentiable in time discrete solutions. A direct connection between the two families by a computationally cheap post-processing is presented. The error estimates are illustrated by numerical experiments.
We propose and study numerically the implicit approximation in time of the Navier-Stokes equations by a Galerkin-collocation method in time combined with inf-sup stable finite element methods in space. The conceptual basis of the Galerkin-collocation
Solutions to the stochastic wave equation on the unit sphere are approximated by spectral methods. Strong, weak, and almost sure convergence rates for the proposed numerical schemes are provided and shown to depend only on the smoothness of the drivi
In this work, we study the reproducing kernel (RK) collocation method for the peridynamic Navier equation. We first apply a linear RK approximation on both displacements and dilatation, then back-substitute dilatation, and solve the peridynamic Navie
A general adaptive refinement strategy for solving linear elliptic partial differential equation with random data is proposed and analysed herein. The adaptive strategy extends the a posteriori error estimation framework introduced by Guignard and No
In this note we develop a fully explicit cut finite element method for the wave equation. The method is based on using a standard leap frog scheme combined with an extension operator that defines the nodal values outside of the domain in terms of the