High‐order space‐time finite element schemes for acoustic and viscodynamic wave equations with temporal decoupling. (7th February 2014)

• Record Type: Journal Article
• Authors: Banks, H. T.; Birch, Malcolm J; Brewin, Mark P; Greenwald, Stephen E; Hu, Shuhua; Kenz, Zackary R; Kruse, Carola; Maischak, Matthias; Shaw, Simon; Whiteman, John R
• Is Part Of: International journal for numerical methods in engineering. Volume 98:Number 2(2014)
• Page Start: 131
• Page End: 156
• Publication Date: 2014-02-07
• Subjects: Numerical analysis -- Periodicals; Engineering mathematics -- Periodicals; 620.001518
• DOI: 10.1002/nme.4631
• Languages: English
• ISSNs: 0029-5981

Abstract

<abstract abstract-type="main" id="nme4631-abs-0001"> <title>SUMMARY</title> <p id="nme4631-para-0001">We revisit a method originally introduced by Werder <italic>et al.</italic> (in Comput. Methods Appl. Mech. Engrg., 190:6685–6708, 2001) for temporally discontinuous Galerkin FEMs applied to a parabolic partial differential equation. In that approach, block systems arise because of the coupling of the spatial systems through inner products of the temporal basis functions. If the spatial finite element space is of dimension <italic>D</italic> and polynomials of degree <italic>r</italic> are used in time, the block system has dimension (<italic>r</italic> + 1)<italic>D</italic> and is usually regarded as being too large when <italic>r</italic> > 1. Werder <italic>et al.</italic> found that the space‐time coupling matrices are diagonalizable over <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4sxtrhv6" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="block" altimg="urn:x-wiley:00295981:media:nme4631:nme4631-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="double-struck">C</mml:mi></mml:math></alternatives> for <italic>r ⩽</italic>100, and this means that the time‐coupled computations within a time step can actually be decoupled. By using either continuous Galerkin or spectral element methods in space, we apply this DG‐in‐time methodology, for the first time, to second‐order wave equations including elastodynamics with and without Kelvin–Voigt and Maxwell–Zener viscoelasticity. An example set of numerical results is given to demonstrate the favourable effect on error and computational work of the moderately high‐order (up to degree 7) temporal and spatio‐temporal approximations, and we also touch on an application of this method to an ambitious problem related to the diagnosis of coronary artery disease. Copyright © 2014 The Authors. <italic>International Journal for Numerical Methods in Engineering</italic> published by John Wiley & Sons Ltd.</p> </abstract>