A Hybridized Discontinuous Galerkin Method with Reduced Stabilization
Tóm tắt
In this paper, we propose a hybridized discontinuous Galerkin (HDG) method with reduced stabilization for the Poisson equation. The reduce stabilization proposed here enables us to use piecewise polynomials of degree
$$k$$
and
$$k-1$$
for the approximations of element and inter-element unknowns, respectively, unlike the standard HDG methods. We provide the error estimates in the energy and
$$L^2$$
norms under the chunkiness condition. In the case of
$$k=1$$
, it can be shown that the proposed method is closely related to the Crouzeix–Raviart nonconforming finite element method. Numerical results are presented to verify the validity of the proposed method.
Tài liệu tham khảo
Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, 2nd edn. Academic Press, Amsterdam (2003)
Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19(4), 742–760 (1982)
Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, New York (2008)
Burman, E., Stamm, B.: Low order discontinuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47(1), 508–533 (2008)
Burman, E., Stamm, B.: Local discontinuous Galerkin method with reduced stabilization for diffusion equations. Commun. Comput. Phys. 5(2–4), 498–514 (2009)
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)
Cockburn, B., Dong, B., Guzmán, J.: A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems. Math. Comput. 77(264), 1887–1916 (2008)
Cockburn, B., Gopalakrishnan, J., Lazarov, R.: Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47, 1319–1365 (2009)
Cockburn, B., Guzmán, J., Soon, S.C., Stolarski, H.K.: An analysis of the embedded discontinuous Galerkin method for second-order elliptic problems. SIAM J. Numer. Anal. 47(4), 2686–2707 (2009)
Cools, R., Mysovskikh, I.P., Schmid, H.J.: Cubature formulae and orthogonal polynomials. J. Comput. Appl. Math. 127(1–2), 121–152 (2001)
Cools, R., Schmid, H.J.: On the (non)-existence of some cubature formulas: gaps between a theory and its applications. J. Complex. 19(3), 403–405 (2003)
Crouzeix, M., Raviart, P.A.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations RAIRO Modél. Math. Anal. Numer. 7, 33–75 (1973)
Kikuchi, F.: Rellich-type discrete compactness for some discontinuous Galerkin FEM. Jpn. J. Ind. Appl. Math. 29, 269–288 (2012)
Lehrenfeld, C.: Hybrid Discontinuous Galerkin Methods for Solving Incompressible Flow Problems. PhD Thesis, RWTH Aachen University (2010)
Lyness, J.N., Cools, R.: A Survey of Numerical Cubature Over Triangles. Proc. Symp. Appl. Math. 48, 127–150 (1994)
Oikawa, I., Kikuchi, F.: Discontinuous Galerkin FEM of hybrid type. JSIAM Lett. 2, 49–52 (2010)
Tong, P.: New displacement hybrid finite element models for solid continua. Int. J. Numer. Meth. Eng. 2, 95–113 (1970)