Everything you always wanted to know about a-posteriori error estimation in finite element methods, but were afraid to ask

In this paper the basic concepts to obtain a posteriori error estimates for the finite element method are reviewed. Explicit residual-based, implicit (namely subdomain and element) residual-based, hierarchical-based, recovery-base and functional-based error estimators as well as goal oriented error estimators are presented for a test elliptic boundary value problem. These notes are an introductory presentation, reviewing in a not-too-technical way the fundamental concepts involved in the subject and do not aim at being exhaustive or complete but rather simple and easy to follow. For more detailed explanations, we refer the interested reader to [3] and eventually to [4],[9],[15],[27],[30], chapter 4 of [37],[38],[39] – and the references therein – where most of the material contained in this report can be found

Strong Stability Preserving Two-Step Runge-Kutta Methods

We investigate the strong stability preserving (SSP) property of two-step Runge– Kutta (TSRK) methods. We prove that all SSP TSRK methods belong to a particularly simple\ud subclass of TSRK methods, in which stages from the previous step are not used. We derive simple order conditions for this subclass. Whereas explicit SSP Runge–Kutta methods have order at most four, we prove that explicit SSP TSRK methods have order at most eight. We present TSRK methods of up to eighth order that were found by numerical search. These methods have larger SSP coefficients than any known methods of the same order of accuracy, and may be implemented in a form with relatively modest storage requirements. The usefulness of the TSRK methods is demonstrated through numerical examples, including integration of very high order WENO discretizations

Monotonicity and Boundedness in general Runge-Kutta methods

UBL - phd migration 201

Monotonicity conditions for multirate and partitioned explicit Runge-Kutta schemes

Multirate schemes for conservation laws or convection-dominated problems seem to come in two ¿avors: schemes that are locally inconsistent, and schemes that lack mass-conservation. In this paper these two defects are discussed for one-dimensional conservation laws. Particular attention will be given to monotonicity properties of the multirate schemes, such as maximum principles and the total variation diminishing (TVD) property. The study of these properties will be done within the framework of partitioned Runge-Kutta methods. It will also be seen that the incompatibility of consistency and mass-conservation holds for ‘genuine’ multirate schemes, but not for general partitioned methods

A numerical study of an adaptive finite element method of lines approach for coupled reaction-diffusion equations in Omega - partialOmega.

A numerical study of an adaptive finite element method of lines (AFEMOL) approach is presented for the approximation of the solution of a system of reaction-diffusion equations coupling species defined on a 2-dimensional domain Ω and species confined to the boundary of the domain ∂Ω. In order to bound the energy norm of the space discretization error, in the AFEMOL the spatial mesh changes automatically at selected times when the underlying triangulation is refined in areas where it is needed. The decision of when and where to modify the mesh is based on the estimation of the space discretization error. The adaptive process and the a-posteriori explicit error estimation exploited in this note are a modification of the pioneer work developed by Bieterman and Babuschka in [Numer. Math. 40 (1982), 339], [Numer. Math. 40 (1982), 373], [J. Comput. Phys. 63 (1986), 33]. The primary interest, in the manuscript, is the effect of the coupling Ω–∂Ω on the performance of the error estimator and the successive adaptive process. Our numerical results indicate that the global error estimators are accurate, the local error indicators are reliable and that the adaptive strategy successfully controls the space discretization error