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Todd J Arbogast

Professor, Core Faculty, Oden Institute
Department of Mathematics, Oden Institute

W. A. "Tex" Moncrief, Jr. Distinguished Professorship in Computational Engineering and Sciences - Applied Mathematics (Holder)

Numerical Analysis, Partial Differential Equations, Subsurface Modeling


Phone: 512-471-0166

Office Location
PMA 11.162

Postal Address
AUSTIN, TX 78712

Ph.D., University of Chicago (1987)
S.M., University of Chicago (1983)
B.S., University of Minnesota (1981)
B.S., University of Minnesota (1981)

Research Interests

Todd Arbogast's areas of expertise include the numerical analysis of partial differential systems, mathematical modeling, and scientific computation. His research includes the development of Eulerian-Lagrangian and weighted essentially non oscillatory (WENO) schemes for advective transport; the study of mixed methods and cell-centered finite differences for nonlinear and geometrically irregular elliptic problems; the modeling and simulation of multi-phase flow through porous media, including fractured and vuggy media, with applications to petroleum production and groundwater resources; numerical homogenization, subgrid upscaling, and domain decomposition of heterogeneous media; and simulation of the dynamics of the Earth's mantle. He is a core member of the Oden Institute for Computational Engineering and Sciences and associate director of the Center for Subsurface Modeling at The University of Texas at Austin, and an associated faculty of the Department of Statistics and Date Sciences.

Curriculum Vita

Available Publications

  1. T. Arbogast, Ch.-S. Huang, and Xikai Zhao. Finite volume WENO schemes for nonlinear parabolic problems with degenerate diffusion on non-uniform meshes. J. Comput. Phys., 399, 2019, to appear. DOI 10.1016/j.jcp.2019.108921.
  2. T. Arbogast and Zhen Tao. A direct mixed–enriched Galerkin method on quadrilaterals for two-phase Darcy flow. Computational Geosci., 23(5):1141–1160, 2019. DOI 10.1007/s10596-019-09871-2.
  3. Shinhoo Kang, T. Bui-Thanh, and T. Arbogast. A hybridized discontinuous Galerkin method for a linear degenerate elliptic equation arising from two-phase mixtures. Comput. Methods Appl. Mech. Engrg., 350:315–336, 2019. DOI 10.1016/j.cma.2019.03.018.
  4. T. O. Quinelato, A. F. D. Loula, M. R. Correa, and T. Arbogast. Full H(div)-approximation of linear elasticity on quadrilateral meshes based on ABF finite elements. Comput. Methods Appl. Mech. Engrg., 347:120–142, 2019. DOI 10.1016/j.cma.2018.12.013.
  5. T. Arbogast and Zhen Tao. Construction of H(div)-conforming mixed finite elements on cuboidal hexahedra. Numer. Math., 142(1):1–32, 2019. DOI 10.1007/s00211-018-0998-7.
  6. Ch.-S. Huang and T. Arbogast. An implicit Eulerian-Lagrangian WENO3 scheme for nonlinear conservation laws. J. Sci. Computing, 77(2):1084–1114, 2018. DOI 10.1007/s10915-018-0738-2.
  7. T. Arbogast, Ch.-S. Huang, and Xikai Zhao. Accuracy of WENO and adaptive order WENO reconstructions for solving conservation laws. SIAM J. Numer. Anal., 56(3):1818–1847, 2018. DOI 10.1137/17M1154758.
  8. T. Arbogast and A. L. Taicher. A cell-centered finite difference method for a degenerate elliptic equation arising from two-phase mixtures. Comput. Geosci., 21(4):701–712, 2017. DOI 10.1007/s10596-017-9649-9.
  9. T. Arbogast, M. A. Hesse, and A. L. Taicher. Mixed methods for two-phase Darcy-Stokes mixtures of partially melted materials with regions of zero porosity. SIAM J. Sci. Comput., 39(2):B375–B402, 2017. DOI 10.1137/16M1091095.
  10. Ch.-S. Huang and T. Arbogast. An Eulerian-Lagrangian WENO scheme for nonlinear conservation laws. Numer. Meth. Partial Diff. Eqns., 33(3):651–680, 2017. DOI 10.1002/num.22091.
  11. T. Arbogast and M.R. Correa. Two families of H(div) mixed finite elements on quadrilaterals of minimal dimension. SIAM J. Numer. Anal., 54(6):3332–3356, 2016. DOI 10.1137/15M1013705.
  12. T. Arbogast and A. L. Taicher. A linear degenerate elliptic equation arising from two-phase mixtures. SIAM J. Numer. Anal., 54(5):3105–3122, 2016. DOI 10.1137/16M1067846.
  13. Ch.-S. Huang, T. Arbogast, and Ch.-H. Hung. A semi-Lagrangian finite difference WENO scheme for scalar nonlinear conservation laws. J. Comput. Phys., 322:559–585, 2016. DOI 10.1016/j.jcp.2016.06.027.
  14. T. Arbogast, D. Estep, B. Sheehan, and S. Tavener. A posteriori error estimates for mixed finite element and finite volume methods for parabolic problems coupled through a boundary. SIAM/ASA J. Uncertainty Quantification, 3:169–198, 2015. DOI 10.1137/140964059.
  15. T. Arbogast and Hailong Xiao. Two-level mortar domain decomposition preconditioners for heterogeneous elliptic problems. Comput. Methods Appl. Mech. Engrg., 292:221–242, 2015. DOI 10.1016/j.cma.2014.10.049.
  16. Ch.-S. Huang, F. Xiao, and T. Arbogast. Fifth order multi-moment WENO schemes for hyperbolic conservation laws. J. Sci. Comput., 64(2):477–507, 2015. DOI 10.1007/s10915-014-9940-z.
  17. Ch.-S. Huang, T. Arbogast, and Ch.-H. Hung. A re-averaged WENO reconstruction and a third order CWENO scheme for hyperbolic conservation laws. J. Comput. Phys., 262:291–312, 2014.
  18. T. Arbogast, D. Estep, B. Sheehan, and S. Tavener. A posteriori error estimates for mixed finite element and finite volume methods for problems coupled through a boundary with non-matching grids. IMA J. Numer. Anal., 34:1625–1653, 2014. DOI 10.1093/imanum/drt049.
  19. T. Arbogast, M. Juntunen, J. Pool, and M. F. Wheeler. A discontinuous Galerkin method for two-phase flow in a porous medium enforcing H(div) velocity and continuous capillary pressure. Comput. Geosci., 17(6):1055–1078, 2013.
  20. T. Arbogast and Hailong Xiao. A multiscale mortar mixed space based on homogenization for heterogeneous elliptic problems. SIAM J. Numer. Anal., 51(1):377–399, 2013.
  21. T. Arbogast, Zhen Tao, and Hailong Xiao. Multiscale mortar mixed methods for heterogeneous elliptic problems. In Jichun Li et al., editors, Recent Advances in Scientific Computing and Applications, volume 586 of Contemporary Mathematics, pages 9–21, Providence, Rhode Island, 2013. Amer. Math. Soc.
  22. T. Arbogast, Ch.-S. Huang, and Ch.-H. Hung. A fully conservative Eulerian-Lagrangian stream-tube method for advection-diffusion problems. SIAM J. Sci. Comput., 34(4):B447–B478, 2012.
  23. T. Arbogast, Ch.-S. Huang, and T. F. Russell. A locally conservative Eulerian-Lagrangian method for a model two-phase flow problem in a one-dimensional porous medium. SIAM J. Sci. Comput., 34(4):A1950–A1974, 2012.
  24. Ch.-S. Huang, T. Arbogast, and Jianxian Qiu. An Eulerian-Lagrangian WENO finite volume scheme for advection problems. J. Comput. Phys., 231(11):4028–4052, 2012. DOI 10.1016/j.jcp.2012.01.030.
  25. T. Arbogast and Wen-Hao Wang. Stability, monotonicity, maximum and minimum principles, and implementation of the volume corrected characteristic method. SIAM J. Sci. Comput., 33(4):1549–1573, 2011.
  26. T. Arbogast. Homogenization-based mixed multiscale finite elements for problems with anisotropy. Multiscale Model. Simul., 9(2):624–653, 2011.
  27. T. Arbogast. Mixed multiscale methods for heterogeneous elliptic problems. In I. G. Graham, Th. Y. Hou, O. Lakkis, and R. Scheichl, editors, Numerical Analysis of Multiscale Problems, volume 83 of Lecture Notes in Computational Science and Engineering, pages 243–283. Springer, 2011.
  28. T. Arbogast and Wenhao Wang. Convergence of a fully conservative volume corrected characteristic method for transport problems. SIAM J. Numer. Anal., 48(3):797–823, 2010.
  29. T. Arbogast and Ch.-S. Huang. A fully conservative Eulerian-Lagrangian method for a convection-diffusion problem in a solenoidal field. J. Comput. Phys., 229(9):3415–3427, 2010. DOI 10.1016/j.jcp.2010.01.009.
  30. Jichun Li, T. Arbogast, and Yunqing Huang. Mixed methods using standard conforming finite elements. Comput. Methods Appl. Mech. Engrg., 198(5):680–692, 2009.
  31. T. Arbogast and M. S. M. Gomez. A discretization and multigrid solver for a Darcy-Stokes system of three-dimensional vuggy porous media. Comput. Geosci., 13(3):331–348, 2009. DOI 10.1007/s10596-008-9121-y.
  32. T. Arbogast and D. S. Brunson. A computational method for approximating a Darcy-Stokes system governing a vuggy porous medium. Comput. Geosci., 11(3):207–218, 2007.
  33. R. Naimi-Tajdar, C. Han, K. Sepehrnoori, T. J. Arbogast, and M. A. Miller. A fully implicit, compositional, parallel simulator for IOR processes in fractured reservoirs. SPE Journal, 12(3), September 2007.
  34. T. Arbogast, G. Pencheva, M. F. Wheeler, and I. Yotov. A multiscale mortar mixed finite element method. Multiscale Model. Simul., 6(1):319–346, 2007.
  35. T. Arbogast, Ch.-S. Huang, and S.-M. Yang. Improved accuracy for alternating-direction methods for parabolic equations based on regular and mixed finite elements. Mathematical Models & Methods in Applied Sciences, 17(8):1279–1305, 2007.
  36. T. Arbogast and Ch.-S. Huang. A fully mass and volume conserving implementation of a characteristic method for transport problems. SIAM J. Sci. Comput., 28(6):2001–2022, 2006.
  37. T. Arbogast and K. J. Boyd. Subgrid upscaling and mixed multiscale finite elements. SIAM J. Numer. Anal., 44(3):1150–1171, 2006.
  38. T. Arbogast and H. L. Lehr. Homogenization of a Darcy-Stokes system modeling vuggy porous media. Comput. Geosci., 10(3):291–302, 2006. 12
  39. T. Arbogast and M. F. Wheeler. A family of rectangular mixed elements with a continuous flux for second order elliptic problems. SIAM J. Numer. Anal., 42:1914–1931, 2005.
  40. T. Arbogast. Analysis of a two-scale, locally conservative subgrid upscaling for elliptic problems. SIAM J. Numer. Anal., 42:576–598, 2004.
  41. T. Arbogast. An overview of subgrid upscaling for elliptic problems in mixed form. In Z. Chen, R. Glowinski, and Kaitai Li, editors, Current Trends in Scientific Computing, volume 329 ofContemporary Mathematics, pages 21–32. American Mathematical Society, 2003.
  42. T. Arbogast and S. L. Bryant. A two-scale numerical subgrid technique for waterflood simulations. SPE J., 7:446–457, Dec. 2002.
  43. T. Arbogast. Implementation of a locally conservative numerical subgrid upscaling scheme for two-phase Darcy flow. Comput. Geosci., 6:453–481, 2002.
  44. T. Arbogast. Numerical subgrid upscaling of two-phase flow in porous media. In Z. Chen, R. E. Ewing, and Z.-C. Shi, editors, Numerical treatment of multiphase flows in porous media, volume 552 of Lecture Notes in Physics, pages 35–49. Springer, Berlin, 2000.
  45. T. Arbogast, L. C. Cowsar, M. F. Wheeler, and I. Yotov. Mixed finite element methods on non-matching multiblock grids. SIAM J. Numer. Anal., 37:1295–1315, 2000.
  46. T. Arbogast, C. N. Dawson, P. T. Keenan, M. F. Wheeler, and I. Yotov. Enhanced cell-centered finite differences for elliptic equations on general geometry. SIAM J. Sci. Comput., 19:404–425, 1998.
  47. T. Arbogast and I. Yotov. A non-mortar mixed finite element method for elliptic problems on non-matching multiblock grids. Comput. Methods Appl. Mech. Engrg., 149:225–265, 1997.
  48. T. Arbogast, M. F. Wheeler, and I. Yotov. Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences. SIAM J. Numer. Anal., 34:828–852, 1997.
  49. T. Arbogast. Computational aspects of dual-porosity models. In U. Hornung, editor, Homogenization and Porous Media, Interdisciplinary Applied Math. Series, pages 203–223. Springer, New York, 1997.
  50. T. Arbogast, S. Bryant, C. Dawson, F. Saaf, Chong Wang, and M. Wheeler. Computational methods for multiphase flow and reactive transport problems arising in subsurface contaminant remediation. J. Comput. Appl. Math., 74:19–32, 1996.
  51. T. Arbogast, M. F. Wheeler, and Nai-Ying Zhang. A nonlinear mixed finite element method for a degenerate parabolic equation arising in flow in porous media. SIAM J. Numer. Anal., 33:1669–1687, 1996.
  52. T. Arbogast, C. N. Dawson, and M. F. Wheeler. A parallel algorithm for two phase multi-component contaminant transport. Applications of Math., 40:163–174, 1995.
  53. T. Arbogast and Zhangxin Chen. On the implementation of mixed methods as nonconforming methods for second order elliptic problems. Math. Comp., 64:943–972, 1995.
  54. T. Arbogast and M. F. Wheeler. A characteristics-mixed finite element method for advection dominated transport problems. SIAM J. Numer. Anal., 32:404–424, 1995.
  55. T. Arbogast. Gravitational forces in dual-porosity systems. II. Computational validation of the homogenized model. Transport in Porous Media, 13:205–220, 1993.
  56. T. Arbogast. Gravitational forces in dual-porosity systems. I. Model derivation by homogenization. Transport in Porous Media, 13:179–203, 1993.
  57. T. Arbogast, M. Obeyesekere, and M. F. Wheeler. Numerical methods for the simulation of flow in root-soil systems. SIAM J. Numer. Anal., 30:1677–1702, 1993.
  58. J. Douglas, Jr., T. Arbogast, P. J. Paes Leme, J. L. Hensley, and N. P. Nunes. Immiscible displacement in vertically fractured reservoirs. Transport in Porous Media, 12:73–106, 1993.
  59. T. Arbogast. The existence of weak solutions to single-porosity and simple dual-porosity models of two-phase incompressible flow. Journal of Nonlinear Analysis: Theory, Methods, and Applications, 19:1009–1031, 1992.
  60. J. Douglas, Jr., J. L. Hensley, and T. Arbogast. A dual-porosity model for waterflooding in naturally fractured reservoirs. Comput. Methods Appl. Mech. Engrg., 87:157–174, 1991.
  61. J. Douglas, Jr. and T. Arbogast. Dual-porosity models for flow in naturally fractured reservoirs. In J. H. Cushman, editor, Dynamics of Fluids in Hierarchical Porous Media, pages 177–221. Academic Press, London, 1990.
  62. T. Arbogast, J. Douglas, Jr., and U. Hornung. Derivation of the double porosity model of single phase flow via homogenization theory. SIAM J. Math. Anal., 21:823–836, 1990.
  63. T. Arbogast and F. A. Milner. A finite difference method for a two-sex model of population dynamics. SIAM J. Numer. Anal., 26:1474–1486, 1989.
  64. T. Arbogast. On the simulation of incompressible, miscible displacement in a naturally fractured petroleum reservoir. R.A.I.R.O. Mod ́el. Math. Anal. Num ́er, 23:5–51, 1989.
  65. T. Arbogast. Analysis of the simulation of single phase flow through a naturally fractured reservoir. SIAM J. Numer. Anal., 26:12–29, 1989.
  • Fellow of the Society for Industrial and Applied Mathematics, 2018
  • Fellow of the American Mathematical Society, 2012
  • Moncrief Grand Challenge Faculty Award, 2012 (The University of Texas at Austin)
  • ICES Distinguished Research Award, 2011 (The University of Texas at Austin)
  • Frank Gerth III Faculty Fellowship, 2008-2012 (The University of Texas at Austin)
  • The President's Associates Centennial Teaching Fellowship in Mathematics, 1997-1998 (TheUniversity of Texas at Austin)
  • National Science Foundation Mathematical Sciences Postdoctoral Research Fellowship, 1989-1992 (University of Houston and Rice University)
  • Robert R. McCormick Fellowship, 1981-1984 (University of Chicago)
  • Sigma Pi Sigma (physics) and Tau Beta Pi (engineering) honor societies
  • Century Fund Scholarship, 1976-1977 (University of Minnesota