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Scholarly Interest Report
Beatrice Riviere
Noah Harding Chair and Professor of Computational and Applied Mathematics
  • Ph.D. Computational and Applied Mathematics (2000) University of Texas at Austin, Austin, TX
  • M.A. Mathematics (1996) Pennsylvania State Universty, State college, PA
  • Licence Mathematics (1993) University Claude Bernard, Lyon, France
  • Diplome d'Ingenieur Engineering and Mathematics (1995) Ecole Centrale de Lyon, Lyon, France
Primary Department
   Department of Computational and Applied Mathematics
Secretary of SIAM Geoscience activity group
AMS member
Member of Interpore
Member of SIAM, AWM
Secretary for SIAM activity group in Geosciences
Member of SIAM, AMS, AWM
 Computational Optimization and Modeling of Porous Media
 Computational Optimization and Modeling of Porous Media
 Home Page
 Scientific computing and numerical analysis
 Home Page
 Scientific computing and numerical analysis
Research Areas
 Numerical Analysis, Scientific Computing, Discontinuous Galerkin Methods, Finite Element Methods, Data Reduction Flow and Transport in Porous Media, Computational Fluid Dynamics, Geomechanics, Mathematical Biology, Blood Flow
Research Interests 19-JAN-2018

My research deals with modeling of complex physical phenomena by partial differential equations (PDEs), and with the development of efficient and accurate numerical methods for the solution of those PDEs. I have formulated and analyzed a class of numerical methods that are of high order, flexible and robust. These methods, commonly referred to as the Discontinuous Galerkin (DG) methods, consist of finite element methods based on discontinuous polynomial approximations. The PDEs arise from applications listed below. I. Flow and Transport in Porous Media Federal and state agencies that are concerned with energy and the environment have an important and common task to address, namely the understanding of flow and transport processes in porous media. A porous medium consists of a solid with a large amount of connected or disconnected holes that allow fluid to pass. Soils and rocks are examples of porous media. In order to better optimize oil and gas production or to strategically define cost-effective contaminant remediation, advanced technologies are essential. Flow and transport in porous media are mathematically modeled by coupled systems of PDEs arising from the balance equations of mass, momentum and energy. The simplest flow model is the single-phase flow (also called Darcy flow) in which the fluid velocity is proportional to some driving force, namely the hydraulic gradient. We proposed and analyzed a local projection of the DG approximation of the Darcy velocity so that a continuous flux was recovered. We also presented superconvergence of the DG method for such problems. In general, subsurface flow is a complex multicomponent multiphase process. We formulated and implemented sequential and coupled DG schemes for solving the two-phase flow problem such as the mixing of oil and water. Our methods were validated with comparisons with either analytical solutions or with more standard numerical approaches (such as the finite volume method). Finally, we derived rigorous a priori error estimates for the coupled DG scheme. The current research on DG methods for multiphase multicomponent flow is still at its early stage. In the case of two-phase single-component flow, I plan to analyze the sequential scheme and simulate more complex problems (discontinuous capillary pressures, three-dimensional domains). I also propose to investigate high order methods for solving two-phase two-component flow, and three-phase flow - both from a theoretical and a computational point of view. II Incompressible Flows: Incompressible flows, described by the Navier-Stokes equations, play an important role in many scientific and industrial applications such as aircraft designs, weather prediction, oil industry and heat exchangers. Numerical methods should handle the coupling of the nonlinear reaction term with the incompressibility condition. Another difficulty is the simulation of turbulent flows with high Reynolds numbers. There exist in the literature many finite element approximations of the steady incompressible Stokes and Navier-Stokes problems. However, there is very little literature on completely discontinuous Galerkin methods. Discontinuous schemes are much easier to implement since no continuity requirement is imposed. We analyzed and implemented a class of discontinuous Galerkin methods for solving the Stokes and Navier-Stokes equations. We first considered the case of domain decomposition where each subdomain was subdivided into a conforming mesh; we then relaxed the condition on the meshes at the interface and allowed for general non-matching meshes. By using an operator splitting technique, we reduced the cost of the method by decoupling the incompressibility condition from the nonlinearity term. In the case of turbulent flows with high Reynolds numbers, Kolmogorov's theory says that accurate direct numerical simulation is not possible even on today's computers. We investigated the subgrid eddy viscosity models for the classical finite element method and the DG method. III Inflammatory Response and Wound Healing: In the U.S., the primary cause of death in critically ill patients is sepsis. Sepsis can be defined as an uncontrolled inflammatory response due to bacteria infection. As of today, there are very few therapeutic options available to patients. Sepsis occurs in particular in the necrotizing enterocolitis (NEC) disease, which is the leading cause of death from gastrointestinal disease in infants less than one month old. NEC results from an injury to the mucosal lining of the intestine, leading to translocation of bacteria and endotoxin. We proposed mathematical models of inflammation and wound healing in the setting of diabetic foot ulcers and in the setting of NEC. In the NEC disease, intestinal mucosal defects are repaired by the process of intestinal restitution during which enterocytes migrate to the sites of injury. Since the usual diffusion model is not valid here, we developed an elasticity-based model of enterocytes migration that incorporated the following components: 1) motility promoting force due to lamellipod formation, 2) motility impeding friction due to the adhesion to the cell matrix and 3) enterocyte proliferation. Our model was calibrated with experimental data. IV Coupling Incompressible Flow and Darcy Flow There is a need for understanding the interaction between incompressible turbulent flow and porous media flow. Examples of such complex flow arise from environmental problems of groundwater contamination through rivers and from energy problems of injection and production wells modeling for gas and oil reservoirs. One of the challenges in solving the coupled problem is to capture the real physical phenomena at the interface between the two fluid regions. We considered three transmissibility conditions: the continuity of fluxes, the balance of forces at the interface and the Beavers-Joseph-Saffman law. We formulated and analyzed discontinuous Galerkin methods and mixed finite element methods for solving the coupled Stokes and Darcy problems. For the coupling of Navier-Stokes and Darcy, we had to modify one interface condition. We then introduced a weak formulation, proved its well-posedness and derived the numerical analysis of a DG scheme. We have generalized our previous results to non-homogeneous boundary conditions and time-dependent flow. We are currently coupling the resulting flow field to a reactive transport equation, as well as with multiphase flow equation. The DG methods are well-suited to these problems, as they conserve mass locally. Applications of such coupling include contamination of groundwater through rivers, or the transport of drugs from blood vessels to organs. VI A Posteriori Error Estimation and Mesh Adaptivity While convergence results establish the theoretical validity of a method, in practice the asymptotic range cannot be reached. Therefore, the ability to refine the mesh locally is essential in controlling the numerical error and in reducing simulation times. Local mesh refinement is driven by the techniques of a posteriori error estimation. Explicit error estimators are computed explicitly from the solution and the given data. They usually can be computed quite efficiently but lead to error indicators containing unknown generic constants. Implicit error estimators attempt to compute guaranteed lower and upper bounds of the error through the solution of a dual problem. We plan to derive a posteriori error estimates for the problems listed above, and to incorporate them in our software.

Teaching Areas
 Numerical Analysis, Numerical Methods for Partial Differential Equations, Finite Element Methods
Selected Publications
 Refereed articles

B. Riviere, J. Tan, T. Thompson "Error Analysis Of Primal Discontinuous Galerkin Methods For A Mixed Formulation Of The Biot Equations." Computers and Mathematics with Applications, 73 (2017) : 666--683.


Beatrice Riviere and Xin Yang "A DG Method for the Simulation of CO2 Storage in Saline Aquifer." Submitted


C. Puelz, B. Riviere "A Priori Error Estimates of Adams-Bashforth Discontinuous Galerkin Methods for Scalar Nonlinear Conservation Laws." Journal of Numerical MathematicsAccepted


C. Puelz, S. Acosta, B. Riviere, D. Penny, K. Brady, C.G. Rusin, "A Computational Study of the Fontan Circulation with Fenestration or Hepatic Vein Exclusion." Computers in Biology and Medicine, 89 (2017) : 405-418.


C. Puelz, S. Canic, B. Riviere, C.G. Rusin "Comparison of Reduced Models for Blood Flow Using Runge Kutta Discontinuous Galerkin Methods." Applied Numerical Mathematics,, 115 (2017) : 114-141.


C. Thiele, M. Araya-Polo, F.O. Alpak, B. Riviere, F. Frank "Inexact hierarchical Scale Separation: a Two-Scale Approach for Linear Systems from Discontinuous Galerkin Discretizations." Computers & Mathematics with ApplicationsAccepted


E. Hendryx, B. Riviere, D. Sorensen, C. Rusin "Finding Representative Electrocardiogram Beat Morphologies with CUR." Journal of Biomedical InformaticsAccepted


F. Frank, C. Liu, A. Scanziani, F.O. Alpak, B. Riviere "An energy-based equilibrium contact angle boundary condition on jagged surfaces for phase-field methods." Submitted


F. Frank, C. Liu, F.O. Alpak, B. Riviere "A finite volume/discontinuous Galerkin method for the advective Cahn-Hilliard equation with degenerate mobility on porous domains stemming from micro-CT imaging." Computational GeosciencesAccepted


M. Fabien, M. Knepley, B. Riviere "Heterogeneous Computing for a Hybridizable Discontinuous Galerkin Geometric Multigrid Method." Submitted


N. Chaabane and B. Riviere "A Sequential Discontinuous Galerkin Method for the Coupling of Flow and Geomechanics." Journal of Scientific ComputingAccepted


N. Chaabane, B. Riviere "A Splitting-Based Finite Element Method for the Biot Poroelasticity System." Computers and Mathematics with ApplicationsAccepted


N. Chaabane, V. Girault, B. Riviere, T. Thompson "A Stable Enriched Galerkin Element for the Stokes Problem." Submitted


N. Chaabane, V. Girault, C. Puelz, B. Riviere "Convergence of IPDG for Coupled Time Dependent Navier-Stokes and Darcy Equations." Journal of Computational and Applied Mathematics, 324 (2017) : 25--48.


S. Acosta, C. Puelz, B. Riviere, D.J. Penny, K.M. Brady, C.G. Rusin "Cardiovascular Mechanics in the Early Stages of Pulmonary Hypertension: a Computational Study." Biomechanics and Modeling in MechanobiologyAccepted


V. Girault, J. Li, B. Riviere "Strong Convergence of the Discontinuous Galerkin Scheme for the Low Regularity Miscible Displacement Equations." Numerical Methods for Partial Differential Equations, 33 (2017) : 489--513.


Vivette Girault, Jizhou Li, Beatrice Riviere "Strong convergence of the discontinuous Galerkin scheme for the low regularity miscible displacement equations." Submitted


J. Li and B. Riviere "High order discontinuous Galerkin method for simulating miscible flooding in porous media." Computational GeosciencesIn Press


J. Li and B. Riviere. "Numerical solutions of the incompressible miscible displacement equations in heterogeneous media", Computer Methods in Applied Mechanics and Engineering, 292, p. 107--121, DOI:10.1016/j.cma.2014.10.048.


J. Li, B. Riviere and N. Walkington. "Convergence of a high order method in time and space for the miscible displacement equations", ESAIM: Mathematical Modelling and Numerical Analysis, 49, p. 953-976, 2015.


J. Young, S. Ozisik, B. Riviere, M. Shamim. "A comprehensive mathematical framework for modeling intestinal smooth muscle cell contraction with applications to intestinal edema", Mathematical Biosciences, 262, p.206--213, 2015.


Justin Dong and Beatrice Riviere "A semi-implicit method for incompressible three-phase flow in porous media." Computational GeosciencesIn Revision


Omer Alpak, Florian Frank, Beatrice Riviere "A phase-field method for the direct simulation of two-phase flows in pore-scale media using a non-equilibrium wetting boundary condition." Computational GeosciencesIn Press


R. Rankin, B. Riviere. "A high order method for solving the black-oil problem in porous media", Advances in Water Resources, 78, p.126--144, 2015.


S. Acosta, C. Puelz, B. Riviere, D. Penny, C. Rusin. "Numerical method of characteristics for one-dimensional blood flow", Journal of Computational Physics, 294, p. 96-109, 2015.


V. Girault, J. Li, B. Riviere "Strong convergence of discrete DG solutions of the heat equation." Journal of Numerical MathematicsIn Revision


B. Riviere and S. Sardar. "Penalty-free discontinuous Galerkin methods for incompressible Navier-Stokes equations", Mathematical Models and Methods in Applied Sciences, 24 (6) p.1217--1236, 2014


A. Cesmelioglu, V. Girault, B. Riviere. "Time-dependent coupling of Navier-Stokes and Darcy flows", ESAIM: Mathematical Modelling and Numerical Analysis, 47 (2), p. 539-554, 2013


B. Riviere. "Discontinuous finite element methods for coupled surface-subsurface flow and transport problems", IMA Volumes in Mathematics and its Applications: Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations, Springer, p. 259-279, 2013.


B. Riviere and S. Sardar. "Penalty-free discontinuous Galerkin methods for incompressible Navier-Stokes equations", Mathematical Models and Methods in Applied Sciences, also technical report TR12-09, to appear, doi:10.1142/S0218202513500826, 2013


B. Riviere and X. Yang. "Convergence Analysis Of A Coupled Method for Time-Dependent Convection-Diffusion Equations", Numerical Methods for Partial Differential Equations, 30 (1), p. 133-157, 2014.


J. Li and B. Riviere and N. Walkington "Convergence of a high order method in time and space for the miscible displacement equations." In Revision


J. Young, B. Riviere, K. Uray, C. Cox. "A mathematical model of intestinal edema formation", Mathematical Medicine & Biology, to appear


K. Liu and B. Riviere. "Discontinuous Galerkin Methods for Elliptic Partial Differential Equations with Random Coefficients", International Journal of Computer Mathematics, 90 (11), p. 2477-2490, 2013


Keyes et al. "Multiphysics Simulations: Challenges and Opportunities", International Journal of High Performance Computing Applications, 27 (1) p.4--83, 2013.


V. Girault, G. Kanschat and B. Riviere "Error analysis for a monolithic discretization of coupled Darcy and Stokes problems." In Press


A. Cesmelioglu and B. Riviere. "Existence of a weak solution for the fully coupled Navier-Stokes/Darcy-Transport problem", Journal of Differential Equations, 252, p. 4138-4175, 2012


P. Chidyagwai and B. Riviere. "A Two-grid method for coupled free flow with porous media flow", Advances in Water Resources, 34 p.1113-1123, 2011.


B. Riviere and N. Walkington. "Convergence of a discontinuous Galerkin method for the miscible displacement under low regularity", SIAM Journal on Numerical Analysis, 49, p. 1085-1110, 2011


P. Chidyagwai, I. Mishev and B. Riviere. "On the coupling of finite volume and discontinuous Galerkin method for elliptic problems", Journal of Computational and Applied Mathematics, 231 p.2193-2204, 2011


A. Cesmelioglu, P. Chidyagwai and B. Riviere ""Continuous and discontinuous finite element methods for coupled surface-subsurface flow and transport problems", submitted, also technical report TR10-09. ." Submitted


G. Kanschat and B. Riviere. "A strongly conservative finite element method for the coupling of Stokes and Darcy flow", Journal of Computational Physics, to appear, doi 10.1016/j.jcp.2010.04.021.


S. Ozisik, B. Riviere and T. Warburton "On the constants in inverse inequalities in L2." Submitted


T. Wihler and B. Riviere.
"Discontinuous Galerkin methods for second-order elliptic PDE with low-regularity solutions",
Journal of Scientific Computing, 46 (2), p. 151-165, 2011


A. Cesmelioglu and B. Riviere. "Analysis of weak solutions for the fully coupled Stokes-Darcy-Transport problem".



A. Cesmelioglu and B. Riviere. "Primal Discontinuous Galerkin Methods For Time-Dependent Coupled Surface And Subsurface Flow", Journal of Scientific Computing, 40, p.115-140, (2009). DOI 10.1007/s10915-009-9274-4


B. Riviere, Y. Epshteyn, D. Swigon and Y. Vodovotz. "A Simple Mathematical Model of Signaling Resulting from the Binding of Lipopolike Receptor 4 Demonstrates Inherent Preconditioning Behavior", Mathematical Biosciences, 217 (1) p. 19-26, 2009 (doi:10.1016/j.mbs.2008.10.002).


J. Guzman and B. Riviere. "Sub-optimal convergence of non-symmetric discontinuous Galerkin methods for odd polynomial approximations", Journal of Scientific Computing, 40, p. 273-280, 2009


J. Proft and B. Riviere. "Discontinuous Galerkin Methods for Convection-Diffusion Equations with varying and vanishing diffusivity", International Journal of Numerical Analysis and Modeling, 6 (4), p.533-561, 2009


V. Girault and B. Riviere. "DG approximation of coupled Navier-Stokes and Darcy equations by Beaver-Joseph-Saffman interface condition", SIAM Journal on Numerical Analysis, 47, p. 2052-2089, 2009


Y. Epshteyn and B. Riviere. "Analysis of hp Discontinuous Galerkin Methods for Incompressible Two-Phase Flow", Journal of Computational and Applied Mathematics, 225 p. 487-509, 2009


Y. Epshteyn, T. Khan and B. Riviere. "Numerical Solution Of A One-Dimensional Inverse Problem By The Discontinuous Galerkin Method", Mathematics and Computers in Simulation, 79 p. 1989-2000, 2009

 Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation. SIAM 2008. ISBN-10: 089871656X
 Refereed conference papers

Beatrice Riviere and Xin Yang "A Domain-Based Multinumeric Method for the Steady-State Convection-Diffusion Equation." In Press


D. E. Keyes, L. C. McInnes, C. Woodward, W. D. Gropp, E. Myra, M. Pernice, J. Bell, J. Brown, A. Clo, J. Connors, E. Constantinescu, D. Estep, K. Evans, C. Farhat, A. Hakim, G. Hammond, G. Hansen, J. Hill, T. Isaac, X. Jiao, K. Jordan, D. Kaushik, E. Kaxiras, A. Koniges, K. Lee, A. Lott, Q. Lu, J. Magerlein, R. Maxwell, M. McCourt, M. Mehl, R. Pawlowski, A.P. Randles , D. Reynolds, B. Riviere, U. Ru ̈de, T. Scheibe, J. Shadid, B. Sheehan, M. Shephard, A. Siegel, B. Smith, X. Tang, C. Wilson, and B. Wohlmuth. "Multiphysics Simulations: Challenges and Opportunities." In Press

 Conference papers

C. Thiele, M. Araya-Polo, F. Alpak, B. Riviere and F. Frank. "Inexact Hierarchical Scale Separation: An Efficient Linear Solver for Discontinuous Galerkin Discretizations.." Proceedings of the Society for Petroleum Engineers conference, SPE-182671-MS (2017)


F. Frank, C. Liu, F. Alpak, M. Araya-Polo and B. Riviere. "A Discontinuous Galerkin Finite Element Framework for the Direct Numerical Simulation of Flow on High-Resolution Pore-Scale Images.." Proceedings of the Society for Petroleum Engineers conference, SPE-182606-MS (2017)


V. Girault, G. Kanschat and B. Riviere. "On the coupling of incompressible Stokes or Navier-Stokes and Darcy flows through porous media". In book "Modelling and Simulation in Fluid Dynamics in Porous Media", ed. Ferreira, Barbeiro, Pena, Wheeler, Conference proceedings, 2012.


Magnhild Lien, Ami Radunskaya, Susanne Brenner and B. Riviere "AWM at the 2017 Annual SIAM Meeting." AWM Newsletter (2017)

 Conference Paper

"High Order Discretization for Simulating Miscible Displacement Process in Porous Media (SIAM Geosciences 2015).." (06/2015)


"Numerical Simulations of the Cahn-Hilliard Equation in Porous Domains (SIAM Geosciences 2015).." (06/2015)

 Conference abstracts

"Hybrid Parallel Implementation of the DG Method."


"Strongly scalable high order algorithm for miscible flooding on massively parallel architecture." Oil and Gas HPC workshop, Rice University. (03/2015) With Jizhou Li


"A Poroelasticity Model for Intestinal Edema." Finite Element Rodeo and Circus, Baton Rouge. (March 2013)

 Invited Talks


  1. A DG Method for Pore Scale Flows, SIAM Geosciences Conference

." Erlangen, Germany. (September 2017)



  1. A Discontinuous Galerkin Finite Element Framework for the Direct Numerical Simulation of Flow

." (February 2017)



  1. Numerical Methods for Coupled Flow and Geomechanics, Worshop on Applied & Computational Math.

." U. Houston. (March 2017)


"A Discontinuous Galerkin Finite Element Framework for the Direct Numerical Simulation of Flow on High-Resolution Pore-Scale Images." (February 2017)


"Pore-scale Thermodynamics and Multiphase Flow Project, Imperial College." (November 2017)


"Numerical Modeling of Viscous Fingering." (April 2016)


"High Order Methods for  Flows in Heterogeneous Porous Media." Advanced Numerical Methods in the Mathematical Sciences, Texas A&M University. (05/2015)


"Numerical Algorithms for coupled free and porous medium flows." Workshop on Coupling of Free and Porous Medium Flow, Stuttgart, Germany. (March 2014)


"Numerical Methods For Flows In Heterogeneous Porous Media." University Pierre et Marie Curie, Paris, France. (May 2014)


"Convergence of High Order Methods for the Miscible Displacement Problem." The Mathematics of Finite Elements and Applications 2013 (MAFELAP), Brunel, England. (June 2013)


"Locally Mass Conservative Methods for Flows in Porous Media." ExxonMobil Upstream Research Company, Houston. (May 2013)


"Locally Mass Conservative Methods with Discontinuous Galerkin in Time for Miscible Displacement in Porous Media." IAM Conference on Mathematical and Computational Issues in the Geosciences, Padova, Italy. (June 2013)


"Numerical solution of miscible displacement under low regularity." SIAM Annual Meeting, San Diego. (July 2013)


"Coupled free flows and porous media flows. 2012 John H. Barrett Memorial Lectures conference, Tennessee.." (05/12)


"Coupling locally mass conservative methods for flow in porous media, {\it Advances in Numerical Analysis \& Scientific Computing}, University of Houston, Houston, (04/11)."


"Multi-numeric methods for porous media flows". Scientific Computing Around Louisiana, New Orleans, January 2011.


"Weak And Numerical Solutions For Coupled Navier-Stokes, Darcy And Transport Equations." Joint Mathematics Meetings, New Orleans. (January 2011)


"Weak Solution and Numerical Solution of Multiphysics Couplings." Department of Mathematics, University of Houston. (February 2010)


"A Multinumerics Method for Solving a Multiphysics Problem." The Mathematics of Finite Elements and Applications, Brunel, England. (June 2009)


"A Weak Solution and A Numerical Solution of the Coupled Navier-Stokes and Darcy Equations." SIAM Conference on Mathematical and Computational Issues in the Geosciences, Leipzig Germany. (June 2009)


"Applications of Discontinuous Galerkin Methods to Complex Flow and Transport Problems." Exxon Mobil Upstream Research, Houston. (February 2009)


"Numerical Methods for Solving the Miscible Displacement Problem." 1051st AMS Meeting, Baylor University. (October 2009)


"Numerical Solution of the Transport of Contaminants in Surface and Subsurface Flows." SIAM Conference on Mathematical and Computational Issues in the Geosciences, Leipzig Germany. (June 2009)


"On the coupled problem of Navier-Stokes, Darcy and transport equations." SIAM Conference on Analysis of Partial Differential Equations, Miami, Florida. (December 2009)


"Multiphysics Couplings in Porous Media." Department of Mathematics, University of Houston. (October 2008)


"Numerical Methods for Solving Coupled Surface and Subsurface Flows." Department of Mathematics, University of Strathclyde, Glasgow, Scotland. (November 2008)

 Keynote Speaker


  1. Numerical Methods for Solving Linear Poroelasticity Equations, AWM Research Symposium 

." UCLA. (April 2017)



  1. Software Development for Porous Media Modeling, Scientific Software Days Conference 

." UT-Austin. (April 2017)


"High Order Methods for Coupled Flow and Transport Problems." 12th U.S. National Congress on Computational Mechanics, Raleigh. (July 2013)


"Discontinuous Galerkin methods for multiphysic problems. 20th International Conference on Domain Decomposition Methods, France.." (06/12)



  1. Discontinuous Galerkin methods for porous media flows and geomechanics, Clemson University.

." (November 2017)


"A DG Method for Pore Scale Flows, Oregon State University,." (October 2017)


"Flexible Numerical Methods for Porous Media Flows." University of Heidelberg, Heidelberg, Germany. (June 2014)


"Flexible Numerical Methods for Reservoir Flows." Institut Francais du Petrole, Paris, France. (May 2014)


"A sequential method for solving the poroelasticy equations,  Finite Element Rodeo Conference." (March 2017)


"PMFS: Technical Review,  Shell." (December 2017)


"Pore-scale thermodynamics and multiphase flow project, Shell." (June 2017)


"Porous Media: Modeling and Simulation, W.D. Von Gonten & Co." (April 2017)


Discontinuous Galerkin Method for Miscible Displacement Simulations, Shell, Houston (04/15).


"Finite Element Rodeo conference: conference organizer."


"Oberwolfach workshop: conference co-organizer."


Invited Panel Member.  "Promoting Diversity at the Graduate Level in Mathematics: a National Forum." Mathematical Sciences Research Institute, UC-Berkeley. (October 2008)



  • One-dimensional model of blood flow discretized with Runge-Kutta discontinuous Galerkin methods 

." IMA. (June 2017)

 Seminar Speaker

"Simulation of Viscous Fingering with High Order Numerical Methods, IFPEN, Paris.." (June 2016)


"Convergence of a High Order Method in Space and in Time for the Miscible Displacement Problem." University of Texas at Austin, Austin, Texas. (October 2014)


"Flexible Discontinuous Galerkin Methods for Complex Flows." Shell company, Houston. (July 2013)


"High Order Flexible Methods for Processes in Porous Media." Geophysical Society of Houstonís Data Processing and Acquis ition seminar, Houston. (November 2013)


"High order methods for reservoir flows." 2014 Rice Oil & Gas HPC Workshop, Houston. (March 2014)


"Reactive Flows in Deformable Complex Media." Mathematics Institute at Oberwolfach , Oberwolfach, Germany. (September 2014)


"A splitting strategy for the coupled surface and subsurface flow problem, {\it Workshop on Splitting and Multiscale Methods for Computational PDEs}, Baylor University, Waco, (09/11)."

Editorial Positions
 Associate Editor, Advances in Water Resources . (2017 - 2917)

 Associate Editor, SIAM Journal on Numerical Analysis . SIAM. (2017 - 2017)

 Member of the Editorial Board, International Journal of Computer Mathematics. (2010 - 2015)

 Member of the Editorial Board, SIAM Journal on Numerical Analysis. (2010 - 2015)

 Member of the Editorial Board, Advances in Water Resources. (2009 - 2015)

Supervised Theses & Dissertations
 Christina Ho, Master Discontinuous Galerkin Formulation for Multi-component Multiphase Flow. (2010) (Thesis Director)

 Prince Chidyagwai, Ph.D. Coupling Surface Flow with Porous Media Flow. (2010) (Thesis Director)

 Xin Wang, Master Discontinuous Galerkin time domain methods for acoustics and comparison with finite difference time domain methods. (2010) (Thesis Committee Member)

 Kun Liu, MS Discontinuous Galerkin Methods for Elliptic Partial Differential Equations with Random Coefficients. (2010) (Thesis Director)

 Aycil Cesmelioglu, Ph.D. Complex Flow and Transport Phenomena in Porous Media. (2010) (Thesis Director)

 Drew Kouri, Master Optimization Governed by Stochastic Partial Differential Equations. (2010) (Thesis Committee Member)

 Tommy Binford, PhD Application of harmonic coordinates to 2D interface problems on regular grids. (2011) (Thesis Committee Member)

 Sevtap Ozisik, Ph.D. Fully computable convergence analysis of discontinuous Galerkin finite element approximation with an arbitrary number of levels of hanging nodes. (2012) (Thesis Director)

 Shirin Sardar, Master Penalty-free discontinuous Galerkin methods for the Stokes and Navier-Stokes equations. (2012) (Thesis Director)

 Xin Wang, Ph.D. Transfer of approximation approaches for subgrid modeling. (2012) (Thesis Committee Member)

 David Trevino Garcia, M.S. Modeling and Interpretation of Magnetic Flux Leakage. (2012) (Thesis Committee Member)

 Yin Huang, M.A. Transparency property of one dimensional acoustic wave equations. (2012) (Thesis Committee Member)

 Xin Yang, M.A. A coupled finite volume and discontinuous Galerkin for convection-diffusion problems. (2012) (Thesis Director)

 Millie Mays, M.A. Discrete search optimization for real-time path planning in satellites. (2012) (Thesis Committee Member)

 Kun Liu, PhD Discontinuous Galerkin Methods for Parabolic Partial Differential Equations with Random Input Data. (2013) (Thesis Director)

 Yenny Chandra, Ph.D. Transient behavior of curved structures. (2013) (Thesis Committee Member)

 Jun Tan, Master Theoretical Convergence of Discontinuous Galerkin Methods for Poroelasticity Equations. (2013) (Thesis Director)

 Kai Gong, PhD Density Functional Theory Study of Microstructure and Phase Behavior of Stimuli-Responsive Polymer Brushes . (2013) (Thesis Committee Member)

 Taylor McNeill, Ph.D. A new filtration of the magnus kernel. (2013) (Thesis Committee Member)

 Nichole Stilwell, M.A. gNek: a GPU accelerated incompressible Navier-Stokes solver. (2013) (Thesis Committee Member)

 Jedidhia Gohlke, M.A. Reduced order modeling in a PDE constrained optimization. (2013) (Thesis Committee Member)

 Beverley Grieshaber, Ph.D. Locking-free discontinuous Galerkin methods for problems in elasticity, using linear and multilinear approximations. (2013) (Thesis Committee Member)

 Jizhou Li, Master Locally Mass-Conservative Method with Discontinuous Galerkin in Time for Solving Miscibl e Displacement Equations under Low regularity. (2013) (Thesis Director)

 Emily Hendryx, M.A. Analysis of ECG signals in heart defects. (2014) (Thesis Co-Director)

 Kai Gong, PhD Molecular modeling of microstructure and thermodynamics of bulk and inhomogeneous complex fluids. (2014) (Thesis Committee Member)

 Muhong Zhou, MA Wave equation based stencil optimizations on a multi-core CPU. (2014) (Thesis Committee Member)

 Christian Goll, Ph.D. Design of Numerical Methods for Simulating Models of a Solid Oxide Fuel Cell. (2014) (Thesis Committee Member)

 Yingpei Wang, Ph.D. Efficient preconditionners for domain decomposition methods. (2014) (Thesis Co-Director)

 Xin Yang, PhD Simulation of CO2 sequestration in saline aquifers using discontinuous Galerkin method. (2014) (Thesis Director)

 Yingpei Wang, Ph.D. On the approximation of the Dirichlet to Neumann map for high contrast two phase composites and its applications to domain decomposition methods. (2014) (Thesis Co-Director)

 David Medina, Master OCCA: a unified approach to multi-threading language. (2014) (Thesis Committee Member)

 Letao Zhang, Ph.D. Deformations of Hilbert Schemes of Points on K3 Surfaces and Representation Theory. (2014) (Thesis Committee Member)

 Emily Hendryx, Master Identifying ECG clusters in congenital heart disease. (2015) (Thesis Co-Director)

 David Medina, Ph.D. OKL: A unified language for parallel architecture. (2015) (Thesis Committee Member)

 Linh Huynh, M.A. Algorithms for optimization problems with simple bound constraints. (2015) (Thesis Committee Member)

 Jizhou Li, Ph.D. Locally conservative methods for reservoir flows. (2015) (Thesis Director)

 Rajesh Gandham, Ph.D. High performance high order numerical methods: applications in ocean modeling. (2015) (Thesis Committee Member)

 Mario Bencomo, Master Discontinuous Galerkin and Finite Difference Methods for the Acoustics Equations with Smooth Coefficients. (2015) (Thesis Committee Member)

 Yin Huang, PhD Linearized extended waveform inversion. (2015) (Thesis Committee Member)

 Rajesh Gandham, Ph.D. High performance high order numerical methods: applications in ocean modeling. (2015) (Thesis Committee Member)

 Jun Tan, Ph.D. Mathematical modeling of intestinal edema. (2015) (Thesis Director)

 Jizhou Li, Ph.D. High order discontinuous Galerkin methods for simulating miscible displacement process in porous media with a focus on minimal regularity. (2015) (Thesis Director)

 Zheng Wang, Master Modeling laser-induced thermotherapy in biological tissue. (2015) (Thesis Committee Member)

 Brianna Lynn, M.A. Optimal Control of Flow and Transport Equations Using Discontinuous Galerkin Methods. (2016) (Thesis Director)

 Jorge Acosta, Ph.D. Holonomy Limits of Cyclic Opers. (2016) (Thesis Committee Member)

 Rujeko Chinomona, M.A. Black oil simulation utilizing a central finite volume scheme. (2016) (Thesis Director)

 Yin Huang, Ph.D. Born Waveform Inversion in Shot Coordinate Domain. (2016) (Thesis Committee Member)

 Charles Puelz, Ph.D. Mathematical modeling of hypoplastic left heart syndrom. (2016) (Thesis Director)

 Chen Liu, Master Pore-scale Simulation of Fluid Flow Using Discontinuous Galerkin Methods. (2016) (Thesis Director)

 Quentin Funk, Ph.D. Two Variants on the Plateau Problem. (2016) (Thesis Committee Member)

 Le Wang, Ph.D. Molecular Simulation and Modeling of Surfactant Systems. (2017) (Thesis Committee Member)

 Mario Bencomo, Ph.D. Discontinuous Galerkin and Finite Difference Methods for the Acoustic Equations with Smooth Coefficients . (2017) (Thesis Committee Member)

 Xiaoqun Mu, Ph.D. Density gradient theory applied in multiphase multi-component system. (2018) (Thesis Committee Member)

 Christopher Nguyen, Ph.D. Direct simulation of processes in porous media. (2018) (Thesis Director)

 Brianna Lynn, Ph.D. Optimization of miscible displacement problems. (2018) (Thesis Director)

 Xiaodi Deng, Ph.D. Parallel-in-time gradient-type method for optimal control problems. (Thesis Committee Member)

 Caleb Magruder, Ph.D. Optimization of PDEs. (Thesis Committee Member)

 Kidist Zeleke, Ph.D. Numerical simulations of surface acoustic wave actuated cell sorting and enantiomer separation. (Thesis Committee Member)

 Charles Conn, Ph.D. The characterization and visualization of multi-phase systems using microfluidic devices. (Thesis Committee Member)

 Xiaodi Deng, Master Well Rate Optimization In Reservoir Simulation. (Thesis Committee Member)

 Caleb Magruder, Ph.D. Projection-based model reduction in the context of optimization with implicit PDE constraints. (Thesis Committee Member)

Positions Held
 Affiliate Member Faculty, McGowan Institute for Regenerative Medicine. (2009 - 2009)

 Secretary, SIAM Activity Group in Geosciences. (2012 - 2014)

 Scientific Visitor, IFPen. (2016 - 2016)

 Affiliate Member Faculty, McGowan Institute for Regenerative Medicine. (2012 - 2012)

 Visiting Professor, University of Heidelberg, Germany. (2014 - 2014)

 Affiliate Member Faculty, McGowan Institute for Regenerative Medicine. (2013 - 2013)