Taiwanese Journal of Mathematics

High-order Discontinuous Galerkin Method for Solving Elliptic Interface Problems

Min-Hung Chen and Rong-Jhao Wu

Full-text: Open access

Abstract

In this study, we develop a high-order accurate discontinuous Galerkin scheme using curvilinear quadrilateral elements for solving elliptic interface problems. To maintain accuracy for curvilinear quadrilateral elements with $Q^k$-polynomial basis functions, we select Legendre-Gauss-Lobatto quadrature rules with $k+2$ integration points, including end points, on each edge. Numerical experiments show quadrature rules are appropriate and the numerical solution converges with the order $k+1$. Moreover, we implement the Uzawa method to rewrite the original linear system into smaller linear systems. The resulting method is three times faster than the original system. We also find that high-order methods are more efficient than low-order methods. Based on this result, we conclude that under the condition of limited computational resources, the best approach for achieving optimal accuracy is to solve a problem using the coarsest mesh and local spaces with the highest degree of polynomials.

Article information

Source
Taiwanese J. Math., Volume 20, Number 5 (2016), 1185-1202.

Dates
First available in Project Euclid: 1 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1498874524

Digital Object Identifier
doi:10.11650/tjm.20.2016.7612

Mathematical Reviews number (MathSciNet)
MR3555896

Zentralblatt MATH identifier
1357.65255

Subjects
Primary: 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods

Keywords
high-order method discontinuous Galerkin method elliptic interface problems curved elements

Citation

Chen, Min-Hung; Wu, Rong-Jhao. High-order Discontinuous Galerkin Method for Solving Elliptic Interface Problems. Taiwanese J. Math. 20 (2016), no. 5, 1185--1202. doi:10.11650/tjm.20.2016.7612. https://projecteuclid.org/euclid.twjm/1498874524


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References

  • D. N. Arnold, F. Brezzi, B. Cockburn and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39 (2002), no. 5, 1749–1779.
  • K. Arrow, L. Hurwicz and H. Uzawa, Studies in Nonlinear Programming, Stanford University Press, Stanford, CA, 1958.
  • I.-L. Chern and Y.-C. Shu, A coupling interface method for elliptic interface problems, J. Comput. Phys. 225 (2007), no. 2, 2138–2174.
  • B. Cockburn, Discontinuous Galerkin methods, ZAMM Z. Angew. Math. Mech. 83 (2003), no. 11, 731–754.
  • B. Cockburn, J. Gopalakrishnan and R. Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal. 47 (2009), no. 2, 1319–1365.
  • B. Cockburn, S. Hou and C.-W. Shu, The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: The multidimensional case, Math. Comp. 54 (1990), no. 190, 545–581.
  • B. Cockburn, S. Y. Lin and C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One-dimensional systems, J. Comput. Phys. 84 (1989), no. 1, 90–113.
  • B. Cockburn and C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: General framework, Math. Comp. 52 (1989), no. 186, 411–435.
  • ––––, The Runge-Kutta local projection $P^1$-discontinuous-Galerkin finite element method for scalar conservation laws, RAIRO Modél. Math. Anal. Numér. 25 (1991), no. 3, 337–361.
  • ––––, The Runge-Kutta discontinuous Galerkin method for conservation laws V: Multidimensional systems, J. Comput. Phys. 141 (1998), no. 2, 199–224.
  • ––––, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal. 35 (1998), no. 6, 2440–2463.
  • ––––, Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput. 16 (2001), no. 3, 173–261.
  • W. J. Gordon and C. A. Hall, Transfinite element methods: Blending-function interpolation over arbitrary curved element domains, Numer. Math. 21 (1973), no. 2, 109–129.
  • G. Guyomarc'h, C.-O. Lee and K. Jeon, A discontinuous Galerkin method for elliptic interface problems with application to electroporation, Comm. Numer. Methods Engrg 25 (2009), no. 10, 991–1008.
  • J. L. Hellrung, L. Wang, E. Sifakis and J. M. Teran, A second order virtual node method for elliptic problems with interfaces and irregular domains in three dimensions, J. Comput. Phys. 231 (2012), no. 4, 2015–2048.
  • R. J. LeVeque and Z. L. Li, The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal. 31 (1994), no. 4, 1019–1044.
  • X.-D. Liu, R. P. Fedkiw and M. Kang, A boundary condition capturing method for Poisson's equation on irregular domains, J. Comput. Phys. 160 (2000), no. 1, 151–178.
  • X.-D. Liu and T. C. Sideris, Convergence of the ghost fluid method for elliptic equations with interface, Math. Comp. 72 (2003), no. 244, 1731–1746.