Some high-order Taylor-models based methods for solving PDEs

We present three different approaches to solving PDEs based Taylor-models. In each approach we stress a high accuracy which can be achieved with a grid of relatively low-order Taylor polynomials. In particular, in solving Dirichlet boundary problem for the Laplace equation with the grid of step h of quadratic Taylor polynomials we get the discretization error of order h^10. Similar results are obtained for evolution equations, with and without time discretization. We study the stability of the resulting schemes and present some numerical examples.