Adaptive finite element methods for hyperbolic systems with application to transient acoustics

Abstract

SUMMARYThe solution of the hyperbolic systems of equations governing transient acoustics by adaptive finite elements and various finite difference time discretization schemes is addressed. Emphasis is placed on the use of a class of implicit Runge-Kutta methods for temporal approximations.

Adaptive hpq finite element methods for the analysis of 3D-based models of complex structures. Part 1. Hierarchical modeling and approximations

“…The subsequent examples of the most prominent textbooks on the topic are those by Szabó and Babuška [71], Schwab [59], and Demkowicz et al [24,26], where the historical background and the extended survey of related bibliography can be found. In the meantime the methods have been applied to various problems of the applied mechanics (compare [17,27,40,43,[45][46][47]56,57,84], for example). Also some extensions of the hp approach to other numerical techniques have appeared.…”

An Error Indicator Monitor Function for an r-Adaptive Finite-Element Method

“…Limited work has been done using a posteriori error estimates in the context of mesh movement for one-dimensional problems [1,8], but to our knowledge, such strategies have not been attempted in higher dimensions. (While our concern is parabolic problems, it is worth noting that global error estimation for hyperbolic problems is also complicated by the combination of local time and space discretization errors [29,30], although in certain cases success in solving the error estimation problem globally is achieved [22]. )…”

A space-time finite element method for the exterior acoustics problem

“…1-5, ͑ii͒ Taylor-Galerkin methods, e.g., Ref. 6, and ͑iii͒ semidiscrete methods which employ standard Galerkin finite element methods in space and classical finite difference techniques for integrating in time ͑also referred to as the method of lines͒, see, e.g., Refs. 7-11.…”