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Scholarly Interest Report
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| Beatrice Riviere | | Associate Professor | | Associate Professor of Computational and Applied Mathematics | | | | e-mail:riviere@rice.edu | | | | Ph.D. Computational and Applied Mathematics (2000) University of Texas at AustinM.A. Mathematics (1996) Pennsylvania State UniverstyLicence Mathematics (1993) University Claude BernardDiplome d'Ingenieur Engineering and Mathematics (1995) Ecole Centrale de Lyon | | | | Primary Department | | Department of Computational and Applied Mathematics | |  | | |
| AMS member | | | Member of the American Mathematical Society | | | | SIAM member | | | Member of the Society for Industrial and Applied Mathematics.
Also member of two activity groups: Geosciences and Computational sciences | | | | AWM member | | | Member of the Association for Women in Mathematics | | | | IMACS member | | | Member of International Association for Mathematics and Computers in Simulations | | |
| Websites | | | Home Page
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| | | Scientific computing and numerical analysis
| | | Scientific computing and numerical analysis
| | | Home Page
| | | Scientific computing and numerical analysis
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| Research Areas | | | Numerical Analysis, Scientific Computing, Discontinuous Galerkin Methods, Finite Element Methods
Flow and Transport in Porous Media, Computational Fluid Dynamics, Mathematical Biology
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| Efficient high order numerical methods for solving partial differential equations | | | 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.
| | | | Efficient high order numerical methods for solving partial differential equations | | | 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 I, Numerical Methods for Partial Differential Equations, Finite Element Methods | | |
| Selected Publications | | | Books | | | Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation. SIAM 2008. ISBN-10: 089871656X | | | | | Refereed Articles | | | A. Cesmelioglu and B. Riviere. "Analysis of time-dependent Navier-Stokes flow coupled with Darcy flow", Journal of Numerical Mathematics, 16 (4) p. 249-280, 2008 | | | | | A. Cesmelioglu and B. Riviere. "Analysis of weak solutions for the fully coupled Stokes-Darcy-Transport problem".
Submitted. | | | | | 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). | | | | | G. Kanschat and B. Riviere. "A strongly conservative finite element method for the coupling of Stokes and Darcy flow", submitted. | | | | | 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 | | | | | Noel Walkington and Beatrice Riviere "Convergence of a discontinuous Galerkin method for the miscible displacement under minimal regularity." In Revision | | | | | 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 | | |
| Presentations | | | Invited Talks | | | "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. (02/13/2009) | | | | | "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) | | | | | Panelist | | | Invited Panel Member. "Promoting Diversity at the Graduate Level in Mathematics: a National Forum." Mathematical Sciences Research Institute, UC-Berkeley. (10/14-10/17) | | | | | Seminar Speaker | | | Invited Speaker. "Multiphysics couplings in porous media." Ronald Hoppe, Department of Mathematics, University of Houston. (10/02/2008) | | | | | Invited Speaker. "Numerical Methods for Solving Coupled Surface and Subsurface Flows." Mark Ainsworth, Department of Mathematics, University of Strathclyde, Glasgow, Scotland. (11/25/2008) | | |
| Positions Held | | | Affiliate Member Faculty, McGowan Institute for Regenerative Medicine. (11/2005 - 11/2011)
| | | Affiliate Member Faculty, McGowan Institute for Regenerative Medicine. (11/2005 - 11/2011)
| | | Affiliate Member Faculty, McGowan Institute for Regenerative Medicine. (11/2005 - 11/2011)
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