Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

If You Use a Screen Reader

This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.

On the Convergence of a Finite Element Method for a Nonlinear Hyperbolic Conservation Law

Claes Johnson and Anders Szepessy
Mathematics of Computation
Vol. 49, No. 180 (Oct., 1987), pp. 427-444
DOI: 10.2307/2008320
Stable URL: http://www.jstor.org/stable/2008320
Page Count: 18
  • Read Online (Free)
  • Download ($34.00)
  • Subscribe ($19.50)
  • Cite this Item
Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
On the Convergence of a Finite Element Method for a Nonlinear Hyperbolic Conservation Law
Preview not available

Abstract

We consider a space-time finite element discretization of a time-dependent nonlinear hyperbolic conservation law in one space dimension (Burgers' equation). The finite element method is higher-order accurate and is a Petrov-Galerkin method based on the so-called streamline diffusion modification of the test functions giving added stability. We first prove that if a sequence of finite element solutions converges boundedly almost everywhere (as the mesh size tends to zero) to a function $u$, then $u$ is an entropy solution of the conservation law. This result may be extended to systems of conservation laws with convex entropy in several dimensions. We then prove, using a compensated compactness result of Murat-Tartar, that if the finite element solutions are uniformly bounded then a subsequence will converge to an entropy solution of Burgers' equation. We also consider a further modification of the test functions giving a method with improved shock capturing. Finally, we present the results of some numerical experiments.

Page Thumbnails

  • Thumbnail: Page 
427
    427
  • Thumbnail: Page 
428
    428
  • Thumbnail: Page 
429
    429
  • Thumbnail: Page 
430
    430
  • Thumbnail: Page 
431
    431
  • Thumbnail: Page 
432
    432
  • Thumbnail: Page 
433
    433
  • Thumbnail: Page 
434
    434
  • Thumbnail: Page 
435
    435
  • Thumbnail: Page 
436
    436
  • Thumbnail: Page 
437
    437
  • Thumbnail: Page 
438
    438
  • Thumbnail: Page 
439
    439
  • Thumbnail: Page 
440
    440
  • Thumbnail: Page 
441
    441
  • Thumbnail: Page 
442
    442
  • Thumbnail: Page 
443
    443
  • Thumbnail: Page 
444
    444