Abstract
We describe an algorithm for optimising the mesh in the least-squares finite element discretisation of first-order systems of partial differential equations. The key feature of the method is that the optimisation process is based entirely on the solution of local PDE problems. We apply the algorithm to the Stokes equations for the flow of a viscous incompressible fluid, and to a convection diffusion equation where convection dominates.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Reference
M. Ainsworth T. Oden (1997) ArticleTitleA posteriori error estimation in finite element analysis Comput. Methods Appl. Mech. Engrg. 142 1–88 Occurrence Handle10.1016/S0045-7825(96)01107-3
M.J. Baines (1994) Moving Finite Elements Clarendon Press Oxford
M.J. Baines (2002) ArticleTitleMoving meshes, conservation laws and least-squares equidistribution Int. J. Numer. Meth. Fluids 40 3–19 Occurrence Handle10.1002/fld.294
R.E. Bank R.K. Smith (1997) ArticleTitleMesh smoothing using a posteriori error estimates SIAM J. Numer. Anal. 34 979–997 Occurrence Handle10.1137/S0036142994265292
Becker R., and Rannacher R., (2001). An optimal control approach to a posteriori error estimation in finite element methods. http://gaia.iwr.uni-heidelberg.de.
P. Bochev M.D. Gunzburger (1994) ArticleTitleAnalysis of least-squares finite element methods for the Stokes equations Math. Comput. 63 479–506
W.L. Briggs (1987) A Multigrid Tutorial SIAM Philadelphia
Z. Cai T.A. Manteuffel S.F. McCormick (1997) ArticleTitleFirst-Order Systems Least- Squares for the Stokes equations, with application to linear elasticity SIAM J. Numer. Anal. 34 1727–1741 Occurrence Handle10.1137/S003614299527299X
Z. Cai C.-O. Lee T.A. Manteuffel S.F. McCormick (2000) ArticleTitleFirst-Order Systems Least-Squares for the Stokes and linear elasticity equations: further results SIAM J. Sci. Comput. 21 1728–1739 Occurrence Handle10.1137/S1064827598338652
J. Deang Gunzburger (1998) ArticleTitleIssues related to finite element methods for the Stokes equations SIAM J. Sci. Comput. 20 878–906 Occurrence Handle10.1137/S1064827595294526
M. Delfour G. Payre J.P. Zolésio (1985) ArticleTitleAn optimal triangulation for secondorder elliptic problems, Comput Methods Appl. Mech. Engrg. 50 231–261 Occurrence Handle10.1016/0045-7825(85)90095-7
J.M. Fiard T.A. Manteuffel S.F. McCormick (1998) ArticleTitleFirst-Order System Least- Squares (FOSLS) for convection-diffusion problems: numerical results SIAM J. Sci. Comput. 19 1958–1979 Occurrence Handle10.1137/S1064827596301169
C.L. Lawson (1977) Software for C1 Interpolation J.R. Rice (Eds) Mathematical Software III Academic Press New York 161–194
R. Li W. Liu H. Ma T. Tang (2002) ArticleTitleAdaptive finite element approximation for distributed elliptic optimal control problems SIAM J. Control Optim. 41 1321–1349 Occurrence Handle10.1137/S0363012901389342
R. Li T. Tang P. Zhang (2002) ArticleTitleA moving mesh finite element algorithm for singular problems in two and three dimensions J. Comput. Phys. 177 365–393 Occurrence Handle10.1006/jcph.2002.7002
P. Roe H. Nishikawa (2002) ArticleTitleAdaptive grid generation by minimizing residuals Int. J. Numer. Meth. Fluids 40 121–136 Occurrence Handle10.1002/fld.336
Y. Tourigny F. Hülsemann (1998) ArticleTitleA new moving mesh algorithm for the finite element solution of variational problems SIAM J. Numer. Anal. 35 1416–1438 Occurrence Handle10.1137/S0036142996313932
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Tourigny, Y. The Optimisation of the Mesh in First-Order Systems Least-Squares Methods. J Sci Comput 24, 219–245 (2005). https://doi.org/10.1007/s10915-004-4614-x
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10915-004-4614-x