Abstract
The BEM-based Finite Element Method is one of the promising strategies which are applicable for the approximation of boundary value problems on general polygonal and polyhedral meshes. The flexibility with respect to meshes arises from the implicit definition of trial functions in a Trefftz-like manner. These functions are treated locally by means of Boundary Element Methods (BEM). The following presentation deals with the formulation of higher order trial functions and their application in uniform and adaptive mesh refinement strategies. For the adaptive refinement, a residual based error estimate is used on general polygonal meshes for the higher order, conforming trial functions. The first numerical results, in the context of adaptive refinement, show optimal rates of convergence with respect to the number of degrees of freedom.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. Marini, A. Russo, Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23(1), 199–214 (2013)
L. Beirão da Veiga, F. Brezzi, L. Marini, Virtual elements for linear elasticity problems. SIAM J. Numer. Anal. 51(2), 794–812 (2013)
L. Beirão da Veiga, K. Lipnikov, G. Manzini, Arbitrary-order nodal mimetic discretizations of elliptic problems on polygonal meshes. SIAM J. Numer. Anal. 49(5), 1737–1760 (2011)
D. Copeland, U. Langer, D. Pusch, From the boundary element domain decomposition methods to local Trefftz finite element methods on polyhedral meshes, in Domain Decomposition Methods in Science and Engineering XVIII. ed. by M. Bercovier, M. Gander, R. Kornhuber, O. Widlund. Lecture Notes in Computational Science and Engineering, vol. 70 (Springer, Berlin/Heidelberg, 2009), pp. 315–322
D.M. Copeland, Boundary-element-based finite element methods for Helmholtz and Maxwell equations on general polyhedral meshes. Int. J. Appl. Math. Comput. Sci. 5(1), 60–73 (2009)
W. Dörfler, A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33(3), 1106–1124 (1996)
Y. Efendiev, J. Galvis, R. Lazarov, S. Weißer, Mixed FEM for second order elliptic problems on polygonal meshes with BEM-based spaces, in Large-Scale Scientific Computing, ed. by I. Lirkov, S. Margenov, J. Waśniewski. Lecture Notes in Computer Science (Springer, Berlin/Heidelberg, 2014), pp. 331–338
R. Hiptmair, A. Moiola, I. Perugia, Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations. Math. Comput. 82(281), 247–268 (2013)
C. Hofreither, L 2 error estimates for a nonstandard finite element method on polyhedral meshes. J. Numer. Math. 19(1), 27–39 (2011)
_, A non-standard finite element method using boundary integral operators, Ph.D. thesis, Johannes Kepler University, Linz, Dec 2012
C. Hofreither, U. Langer, C. Pechstein, Analysis of a non-standard finite element method based on boundary integral operators. Electron. Trans. Numer. Anal. 37, 413–436 (2010)
_____________________________, A non-standard finite element method for convection-diffusion-reaction problems on polyhedral meshes. AIP Conf. Proc. 1404(1), 397–404 (2011)
P. Joshi, M. Meyer, T. DeRose, B. Green, T. Sanocki, Harmonic coordinates for character articulation. ACM Trans. Graph. 26(3), 71.1–71.9 (2007)
S. Martin, P. Kaufmann, M. Botsch, M. Wicke, M. Gross, Polyhedral finite elements using harmonic basis functions. Comput. Graph. Forum 27(5), 1521–1529 (2008)
W.C.H. McLean, Strongly Elliptic Systems and Boundary Integral Equations (Cambridge University Press, Cambridge, 2000)
C. Pechstein, C. Hofreither, A rigorous error analysis of coupled FEM-BEM problems with arbitrary many subdomains. Lect. Notes Appl. Comput. Mech. 66, 109–132 (2013)
S. Rjasanow, O. Steinbach, The Fast Solution of Boundary Integral Equations. Mathematical and Analytical Techniques with Applications to Engineering (Springer, New York/London, 2007)
S. Rjasanow, S. Weißer, Higher order BEM-based FEM on polygonal meshes. SIAM J. Numer. Anal. 50(5), 2357–2378 (2012)
_____, FEM with Trefftz trial functions on polyhedral elements. J. Comput. Appl. Math. 263, 202–217 (2014)
A. Tabarraei, N. Sukumar, Application of polygonal finite elements in linear elasticity. Int. J. Comput. Methods 3(4), 503–520 (2006)
S. Weißer, Residual error estimate for BEM-based FEM on polygonal meshes. Numer. Math. 118(4), 765–788 (2011)
____________________________, Finite Element Methods with local Trefftz trial functions, Ph.D. thesis, Universität des Saarlandes, Saarbrücken, Sept 2012
_______________________________________________________________________, Arbitrary order Trefftz-like basis functions on polygonal meshes and realization in BEM-based FEM. Comput. Math. Appl. 67(7), 1390–1406 (2014)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Weißer, S. (2015). Residual Based Error Estimate for Higher Order Trefftz-Like Trial Functions on Adaptively Refined Polygonal Meshes. In: Abdulle, A., Deparis, S., Kressner, D., Nobile, F., Picasso, M. (eds) Numerical Mathematics and Advanced Applications - ENUMATH 2013. Lecture Notes in Computational Science and Engineering, vol 103. Springer, Cham. https://doi.org/10.1007/978-3-319-10705-9_23
Download citation
DOI: https://doi.org/10.1007/978-3-319-10705-9_23
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-10704-2
Online ISBN: 978-3-319-10705-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)