Optimal order a posteriori error estimates for a class of Runge-Kutta and Galerkin methods
The main purpose of the paper is to introduce a new methodology for performing a posteriori error estimates, which exhibit optimal global order, for a class of time stepping methods of any order that include Runge-Kutta collocation (RK-C) methods and the continuous Galerkin (cG) method for linear and nonlinear stiff ordinary differential equations and parabolic partial differential equations. The first section is an introduction in nature. The second section presents an abstract framework for the discretization and time reconstruction, with emphasis on the simple linear case. These results are applied to cG and then extended to nonlinear equations in the third section. The fourth section deals with RK-C. The authors discuss the relation between the classical order and the stage order. Thus, viewing RK-C methods as collocation or Galerkin-type methods, they clarify the connection between the stage order and the order of convergence in \(L^\infty([0,T];H)\). The latter is \(O(k^{q+1})\) because the approximate solution is a piecewise polynomial of degree \(q\). In the last section one provides explicit conditions on the data and the reconstruction which guarantee the validity for time discrete case.
scientific article; zbMATH DE number 5674531
scientific article