Plane wave discontinuous Galerkin methods: Analysis of the h-version
Claude J. Gittelson; Ralf Hiptmair; Ilaria Perugia
ESAIM: Mathematical Modelling and Numerical Analysis (2009)
- Volume: 43, Issue: 2, page 297-331
- ISSN: 0764-583X
Access Full Article
topAbstract
topHow to cite
topReferences
top- M. Ainsworth, Discrete dispersion relation for hp-version finite element approximation at high wave number. SIAM J. Numer. Anal.42 (2004) 563–575.
- D. Arnold, F. Brezzi, B. Cockburn and L. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal.39 (2002) 1749–1779.
- I. Babuška and J. Melenk, The partition of unity method. Int. J. Numer. Methods Eng.40 (1997) 727–758.
- I. Babuška and S. Sauter, Is the pollution effect of the FEM avoidable for the Helmholtz equation? SIAM Review42 (2000) 451–484.
- L. Banjai and S. Sauter, A refined Galerkin error and stability analysis for highly indefinite variational problems. Report 03-06, Institut für Mathematik, Universität Zürich, Zürich, Switzerland (2006).
- S. Brenner and R. Scott, Mathematical theory of finite element methods, Texts in Applied Mathematics. Springer-Verlag, New York, 2nd edn. (2002).
- A. Buffa and P. Monk, Error estimates for the ultra weak variational formulation of the Helmholtz equation. ESAIM: M2AN42 (2008) 925–940.
- P. Castillo, B. Cockburn, I. Perugia and D. Schötzau, An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal.38 (2000) 1676–1706.
- O. Cessenat, Application d'une nouvelle formulation variationnelle aux équations d'ondes harmoniques. Ph.D. Thesis, Université Parix IX Dauphine, Paris, France (1996).
- O. Cessenat and B. Després, Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz equation. SIAM J. Numer. Anal.35 (1998) 255–299.
- O. Cessenat and B. Després, Using plane waves as base functions for solving time harmonic equations with the ultra weak variational formulation. J. Comp. Acoust.11 (2003) 227–238.
- P. Cummings and X.-B. Feng, Sharp regularity coefficient estimates for complex-valued acoustic and elastic Helmholtz equations. Math. Models Methods Appl. Sci.16 (2006) 139–160.
- B. Despres, Sur une formulation variationnelle de type ultra-faible. C. R. Acad. Sci. Paris, Ser. I318 (1994) 939–944.
- C. Farhat, I. Harari and U. Hetmaniuk, A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime. Comput. Methods Appl. Mech. Eng.192 (2003) 1389–1419.
- C. Farhat, R. Tezaur and P. Weidemann-Goiran, Higher-order extensions of a discontinuous Galerkin method for mid-frequency Helmholtz problems. Int. J. Numer. Meth. Engr.61 (2004) 1938–1956.
- G. Gabard, Discontinuous Galerkin methods with plane waves for the displacement-based acoustic equation. Int. J. Numer. Meth. Engr.66 (2006) 549–569.
- G. Gabard, Discontinuous Galerkin methods with plane waves for time-harmonic problems. J. Comp. Phys.225 (2007) 1961–1984.
- C. Gittelson, R. Hiptmair and I. Perugia, Plane wave discontinuous Galerkin methods. Preprint NI07088-HOP, Isaac Newton Institute Cambride, Cambrid, UK (2007). Available at . URIhttp://www.newton.cam.ac.uk/preprints/NI07088.pdf
- U. Hetmaniuk, Stability estimates for a class of Helmholtz problems. Communications in Mathematical Sciences5 (2007) 665–678.
- R. Hiptmair and P. Ledger, A quadrilateral edge element scheme with minimum dispersion. Report 2003-17, SAM, ETH Zürich, Zürich, Switzerland (2003).
- T. Huttunen and P. Monk, The use of plane waves to approximate wave propagation in anisotropic media. J. Comput. Math.25 (2007) 350–367.
- T. Huttunen, P. Monk and J. Kaipio, Computational aspects of the ultra-weak variational formulation. J. Comp. Phys.182 (2002) 27–46.
- T. Huttunen, M. Malinen and P. Monk, Solving Maxwell's equations using the ultra weak variational formulation. J. Comp. Phys.223 (2007) 731–758.
- F. Ihlenburg, Finite Element Analysis of Acoustic Scattering, Applied Mathematical Sciences132. Springer-Verlag, New York (1998).
- O. Laghrouche, P. Bettes and R. Astley, Modelling of short wave diffraction problems using approximating systems of plane waves. Int. J. Numer. Meth. Engr.54 (2002) 1501–1533.
- J. Melenk, On Generalized Finite Element Methods. Ph.D. Thesis, University of Maryland, USA (1995).
- P. Monk and D. Wang, A least squares method for the Helmholtz equation. Comput. Methods Appl. Mech. Eng.175 (1999) 121–136.
- E. Perrey-Debain, O. Laghrouche and P. Bettess, Plane-wave basis finite elements and boundary elements for three-dimensional wave scattering. Phil. Trans. R. Soc. London A362 (2004) 561–577.
- H. Riou, P. Ladevéze and B. Sourcis, The multiscale VTCR approach applied to acoustics problems. J. Comp. Acoust. (2008) (to appear).
- A. Schatz, An observation concerning Ritz-Galerkin methods with indefinite bilinear forms. Math. Comp.28 (1974) 959–962.
- C. Schwab, p- and hp-Finite Element Methods. Theory and Applications in Solid and Fluid Mechanics, Numerical Mathematics and Scientific Computation. Clarendon Press, Oxford (1998).
- M. Stojek, Least-squares Trefftz-type elements for the Helmholtz equation. Int. J. Numer. Meth. Engr.41 (1998) 831–849.
- R. Tezaur and C. Farhat, Three-dimensional discontinuous Galerkin elements with plane waves and lagrange multipliers for the solution of mid-frequency Helmholtz problems. Int. J. Numer. Meth. Engr.66 (2006) 796–815.