Galerkin and Runge–Kutta methods: unified formulation, a posteriori error estimates and nodal superconvergence

Galerkin and Runge-Kutta methods: unified formulation, a posteriori error estimates and nodal superconvergence

We unify the formulation and analysis of Galerkin and Runge–Kutta methods for the time discretization of parabolic equations. This, together with the concept of reconstruction of the approximate solutions, allows us to establish a posteriori superconvergence estimates for the error at the nodes for all methods.

Optimal order a posteriori error estimates for a class of Runge-Kutta and Galerkin methods

We derive 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 ODEs and parabolic PDEs. The key ingredients in deriving these bounds are appropriate one-degree higher continuous reconstructions of the approximat...

A Posteriori Error Estimates and Mesh Adaptation Strategy for Discontinuous Galerkin Methods Applied to Diffusion Problems

Unified primal formulation-based a priori and a posteriori error analysis of mixed finite element methods

We derive in this paper a unified framework for a priori and a posteriori error analysis of mixed finite element discretizations of second-order elliptic problems. It is based on the classical primal weak formulation, the postprocessing of the potential proposed in [T. Arbogast and Z. Chen, On the implementation of mixed methods as nonconforming methods for secondorder elliptic problems, Math. ...

A posteriori error estimates for discontinuous Galerkin methods for the generalized Korteweg-de Vries equation

with periodic boundary conditions on the interval [0, 1] where ✏ is a positive parameter. The particular equations in (1) are part of a more general class which have arisen in recent years as, e.g., approximate models for the unidirectional propagation of waves in a variety of nonlinear, dispersive media, cf. [8, 24, 9, 10]. The equation in (1) can be seen as an important case of dispersive app...