A class of discontinuous Petrov–Galerkin methods. Part III: Adaptivity

A Class of Discontinuous Petrov-galerkin Methods. Part I: the Transport Equation

Considering a simple model transport problem, we present a new finite element method. While the new method fits in the class of discontinuous Galerkin (DG) methods, it differs from standard DG and streamline diffusion methods, in that it uses a space of discontinuous trial functions tailored for stability. The new method, unlike the older approaches, yields optimal estimates for the primal vari...

A Class of Discontinuous Petrov-galerkin Methods. Part Ii: Optimal Test Functions

We lay out a program for constructing discontinuous Petrov-Galerkin (DPG) schemes having test function spaces that are automatically computable to guarantee stability. Given a trial space, a DPG discretization using its optimal test space counterpart inherits stability from the well-posedness of the undiscretized problem. Although the question of stable test space choice had attracted the atten...

A Class of Discontinuous Petrov-Galerkin Methods. Part IV: Wave Propagation

The phase error, or the pollution effect in the finite element solution of wave propagation problems, is a well known phenomenon that must be confronted when solving problems in the high-frequency range. This paper presents a new method with no phase errors for onedimensional time-harmonic wave propagation problems. The method is constructed within the framework of the Discontinuous Petrov-Gale...

Discontinuous Galerkin Methods for Friedrichs' Systems. Part III. Multifield Theories with Partial Coercivity

This paper is the third and last part of a work attempting to give a unified analysis of Discontinuous Galerkin methods. The purpose of this paper is to extend the framework that has been developed in part II for two-field Friedrichs’ systems associated with second-order PDE’s. We now consider two-field Friedrichs’ systems with partial L2-coercivity and three-field Friedrichs’ systems with an e...

Class Notes Part III – Memory models