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Book

Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation.
SIAM 2008.
ISBN-10: 089871656X.
ISBN-13: 978-0-898716-56-6
Available at SIAM. Part of SIAM Bestsellers.

Refereed Publications (Revues avec comite de lecture):

1. 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, also technical report TR10-23: pdf.

   This paper analyzes the surface/subsurface flow coupled with transport. The flow is modeled by the coupling of Navier–Stokes and Darcy equations. The transport of a species is modeled by a convection-dominated parabolic equation. The two-way coupling between flow and transport is nonlinear and it is done via the velocity field and the viscosity. This problem arises from a variety of natural phenomena such as the contamination of the groundwater through rivers. The main result is existence and stability bounds of a weak solution.

2. 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, also technical report TR10-21. doi:10.1016/j.advwatres.2011.04.010

   This paper presents a two-grid method for solving systems of partial differential equations modelling incompressible free flow coupled with porous media flow. This work considers both the coupled Stokes and Darcy as well as the coupled Navier–Stokes and Darcy problems. The numerical schemes proposed are based on combinations of the continuous finite element method and the discontinuous Galerkin method. Numerical errors and convergence rates for solutions obtained from the two-grid method are presented. CPU times for the two-grid algorithm are shown to be significantly less than those obtained by solving the fully coupled problem.

3. 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, also technical report TR09-18: pdf.

   Discontinuous Galerkin time discretizations are combined with the mixed finite element and continuous finite element methods to solve the miscible displacement problem. Stable schemes of arbitrary order in space and time are obtained. Under low regularity assumptions on the data, convergence of the scheme is proved by using compactness results for functions that may be discontinuous in time.

4. 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, doi:10.1016/j.cam.2010.10.017, also technical report TR10-10: pdf.

   The coupling of cell-centered finite volume method with primal discontinuous Galerkin method is introduced in this paper for elliptic problems. Convergence of the method with respect to the mesh size is proved. Numerical examples confirm the theoretical rates of convergence. Advantages of the coupled scheme are shown for problems with discontinuous coefficients or anisotropic diffusion matrix.

5. 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.

   In this paper we derive an a priori error analysis for interior penalty discontinuous Galerkin finite element discretizations of the Poisson equation with exact solution in $W^{2,p}$, $p\in(1,2]$. We show that the DGFEM converges at an optimal algebraic rate with respect to the number of degrees of freedom.

6. G. Kanschat and B. Riviere. "A strongly conservative finite element method for the coupling of Stokes and Darcy flow", Journal of Computational Physics, 229 (17), p. 5933-5943, 2010, doi 10.1016/j.jcp.2010.04.021.

   We consider a model of coupled free and porous media flow governed by Stokes and Darcy equations with the Beavers-Joseph-Saffman interface condition. This model is discretized using divergence conforming finite elements for the velocities in the whole domain. Discontinuous Galerkin techniques and mixed methods are used in the Stokes and Darcy subdomains, respectively. This discretization is strongly conservative in Hdiv(Omega) and we show convergence. Numerical results validate our findings and indicate optimal convergence orders.

7. P. Chidyagwai and B. Riviere. "Numerical Modelling of Coupled Surface and Subsurface Flow Systems", Advances in Water Resources, 33, p.92-105, 2010 : doi:10.1016/j.advwatres.2009.10.012, pdf.

   This paper presents and compares several numerical solutions of the coupled Navier-Stokes and Darcy equations. The schemes are based on combinations of the finite element method and the discontinuous Galerkin method. Accuracy and robustness of the methods are investigated for heterogeneous porous media. The importance of local mass conservation for filtration problems is also discussed.

8. 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: pdf

   This work formulates and analyzes a new family of discontinuous Galerkin methods for the time-dependent convection-diffusion equation with highly varying diffusion coefficients, that do not require the use of slope limiting techniques. The proposed methods are based on the standard NIPG/SIPG techniques, but use special diffusive numerical fluxes at some important interfaces. The resulting numerical solutions have an L2 error that is significantly smaller than the error obtained with standard discontinuous Galerkin methods. Theoretical convergence results are also obtained.

9. P. Chidyagwai and B. Riviere. "On the solution of the coupled Navier-Stokes and Darcy equations", Computer Methods in Applied Mechanics and Engineering, 198, p. 3806-3820, 2009, (doi:10.1016/j.cma.2009.08.012), pdf.

   This paper introduces and analyzes two models coupling the incompressible Navier-Stokes equations with the porous media flow equations. A numerical method that uses continuous finite elements in the porous medium is proposed. Existence and uniqueness results under small data condition of the numerical solution are proved. Optimal a priori error estimates are derived. Numerical results comparing the two models are provided.

10. 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, (also technical report TR-MATH 07-09 2007).

    In this work, we couple the incompressible steady Navier–Stokes equations with the Darcy equations, by means of the Beavers–Joseph–Saffman's condition on the interface. Under suitable smallness conditions on the data, we prove existence of a weak solution as well as some a priori estimates. We establish local uniqueness when the data satisfy additional smallness restrictions. Then we propose a discontinuous Galerkin scheme for discretizing the equations and do its numerical analysis.

11. 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).

    This paper introduces and analyzes a numerical method based on discontinuous finite element methods for solving the two-dimensional coupled problem of time-dependent incompressible Navier-Stokes equations with the Darcy equations through Beavers-Joseph-Saffman's condition on the interface. The proposed method employs Crank-Nicolson discretization in time (which requires one step of a first order scheme namely backward Euler) and primal DG method in space. With the correct assumption on the first time step optimal error estimates are obtained that are high order in space and second order in time.

12. 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 (doi 10.1007/s10915-008-9255-z).

    We numerically verify that the non-symmetric interior penalty Galerkin method and the Oden-Babuska-Baumann method have sub-optimal convergence properties when measured in the L2-norm for odd polynomial approximations. We provide numerical examples that use piece-wise linear and cubic polynomials to approximate a second-order elliptic problem in one and two dimensions.

13. 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, (doi:10.1016/j.matcom.2008.08.002)

    In this paper, we combine a Tikhonov regularization with a discontinuous Galerkin method to solve an inverse problem in one-dimension. We show that the regularization is simpler than in the case of the inversion using continuous finite elements. We numerically demonstrate that there exist optimal step sizes and polynomial degrees for inversion using the DG method. Numerical results are compared with those obtained by applying the standard finite element method with B-splines as a basis.

14. 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 (also technical report TR-MATH 06-17 2006).

    In this paper, we prove the convergence of a class of discontinuous Galerkin methods for solving the fully coupled incompressible two-phase flow problem in the non-degenerate case. Estimates in both the mesh size and the polynomial degrees are obtained. Numerical convergence rates confirm the theoretical results.

15. 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).

16. 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.
17. Y. Epshteyn and B. Riviere. "Convergence of high order methods for miscible displacement", International Journal of Numerical Analysis and Modeling, 5 p.47-63, 2008, pdf.

    We derive error estimates for a fully discrete scheme using primal discontinuous Galerkin discretization in space and backward Euler discretization in time. The estimates in the energy norm are optimal with respect to the mesh size and suboptimal with respect to the polynomial degree. The proposed scheme is of high order as polynomial approximations of pressure and concentration can take any degree. In addition, the method can handle different types of boundary conditions and is well-suited for unstructured meshes.

18. Q. Mi, D. Swigon, B. Riviere, S. Cetin, Y. Vodovotz, D. Hackam. "One-dimensional elastic continuum model of enterocyte layer migration", Biophysical Journal, 93, p. 3745-3752, 2007.
19. Q. Mi, B. Riviere, G. Clermont, D.L. Steed, Y. Vodovotz, "Agent-based modeling of inflammation and wound healing: insights into diabetic foot ulcer pathology and the role of transforming growth factor-beta 1", Wound Repair and Regeneration, to appear (2007), pdf.

    Inflammation and wound healing are inextricably linked and complex processes, and are deranged in the setting of chronic, non-healing diabetic foot ulcers (DFU). An ideal therapy for DFU should both suppress excessive inflammation while enhancing healing. We reasoned that biological simulation would clarify mechanisms ..

20. B. Riviere, S. Shaw and J.R. Whiteman. "Discontinuous Galerkin finite element methods for dynamic linear solid viscoelasticity", NMPDE, to appear, 2006.
21. Y. Epshteyn and B. Riviere. "Estimation of penalty parameters for symmetric interior penalty Galerkin methods", Journal of Computational and Applied Mathematics, to appear 2006.

    This paper presents computable lower bounds of the penalty parameters for stable and convergent symmetric interior penalty Galerkin methods. In particular, we derive the explicit dependence of the coercivity constants with respect to the polynomial degree and the angles of the mesh elements. Numerical examples in all dimensions and for different polynomial degrees are presented. We show the numerical effects of loss of coercivity.

22. B. Riviere and S. Shaw. "Discontinuous Galerkin finite element approximation of nonlinear non-Fickian diffusion in viscoelastic polymers", SIAM Journal on Numerical Analysis, to appear, 2006, also technical report BICOM 04/03.

    We consider discrete schemes for a nonlinear model of non-Fickian diffusion in viscoelastic polymers. The model is motivated by, but not the same as, that proposed by Cohen et al. in SIAM J.~Appl.~Math., 55, pp.~348--368, 1995. The spatial discretization is effected with both the symmetric and non-symmetric interior penalty discontinuous Galerkin finite element method, and the time discretisation is of Crank-Nicolson type. We also discuss two means of handling the nonlinearity: either implicitly, which requires the solution of nonlinear equations at each time level, or through a linearisation based on extrapolating from previous time levels. The same optimal orders of convergence are proven in both cases and numerical results are also given. These results indicate that the model is capable of capturing physical effects that have been experimentally observed.

23. W. Klieber and B. Riviere. "Adaptive simulations of two-phase flow by discontinuous Galerkin methods", Computer Methods in Applied Mechanics and Engineering, to appear, 2006.

    In this paper we present and compare primal discontinuous Galerkin formulations of the two-phase flow equations. The wetting phase pressure and saturation equations are decoupled and solved sequentially. Proposed adaptivity in space and time techniques yield accurate and efficient solutions. Slope limiters valid on nonconforming meshes are also presented. Numerical examples of homogeneous and heterogeneous media are considered.

24. Y. Epshteyn and B. Riviere. "Fully Implicit Discontinuous Finite Element Methods for Two-Phase Flow", Applied Numerical Mathematics, to appear, 2006.

    In this paper we present two schemes based on discontinuous Galerkin methods for modeling fully implicit formulations of two-phase flow problems arising in porous media. Convergence with respect to uniform mesh refinement or increase in the polynomial degree are considered. Compared to sequential discontinuous schemes, our proposed schemes do not require slope limiting or upwind stabilization techniques. Numerical examples of homogeneous and heterogeneous media on structured and unstructured meshes show the robustness of the method.

25. S. Kaya, W. Layton and B. Riviere. "Subgrid Stabilized Defect Correction Methods for the Navier-Stokes Equations", SIAM Journal on Numerical Analysis, to appear, 2006.

    We consider the synthesis of a recent subgrid stabilization method with defect correction methods. The combination is particularly efficient and combines the best algorithmic features of each. We prove convergence of the method for a fixed number of corrections as the mesh size goes to zero, and derive parameter scalings from the analysis. We also present some numerical tests which both verify the theoretical predictions and illustrate the methods promise.

26. Y. Epshteyn and B. Riviere. "On the solution of incompressible two-phase flow by a p-version discontinuous Galerkin method", Communications in Numerical Methods in Engineering, 22 p.741-751, 2006, pdf.

    This paper presents a fully implicit scheme for approximating two-phase flow in heterogeneous porous media. The primary unknowns are the wetting phase pressure and non-wetting phase saturation. At each time step, a jacobian matrix is computed. Convergence of the scheme is shown via increase of the polynomial degree. No slope limiters are needed.

27. V. Girault, B. Riviere and M. Wheeler. "A splitting method using discontinuous Galerkin for the transient incompressible Navier-Stokes equations", Mathematical Modelling and Numerical Analysis (M2AN), 39 no 6, p. 1115-1148, 2005, also TR-MATH 04-08.

    In this paper we solve the time-dependent incompressible Navier-Stokes equations by splitting the non-linearity and incompressibility, and using discontinuous or continuous finite element methods in space. We prove optimal error estimates for the velocity and suboptimal estimates for the pressure. We present some numerical experiments.

28. B. Riviere and V. Girault. "Discontinuous finite element methods for incompressible flows on subdomains with non-matching interfaces", Computer Methods in Applied Mechanics and Engineering, 195 p.3274-3292, 2006, online.

    In this paper, an improved inf-sup condition is derived for a class of discontinuous Galerkin methods for solving the steady-state incompressible Stokes and Navier-Stokes equations. The computational domain is subdivided into subdomains with non-matching meshes at the interfaces. Optimal error estimates are obtained. Numerical experiments including two benchmark problems are presented.

29. S. Kaya and B. Riviere. "A two-grid stabilization method for solving the steady-state Navier-Stokes equations", Numerical Methods for Partial Differential Equations, 22 no 3, p. 728-743, 2006, also TR-MATH 04-06, pdf.

    We formulate a subgrid eddy viscosity method for solving the steady-state incompressible flow problem. The eddy viscosity does not act on the large flow structures. Optimal error estimates are obtained for velocity and pressure. The numerical illustrations agree completely with the theoretical results.

30. S. Kaya and B. Riviere. "A discontinuous subgrid eddy viscosity method for the time-dependent Navier-Stokes equations", SIAM Journal on Numerical Analysis, 43 no 4, p. 1572-1595, 2005, also technical report TR-MATH 03-14: pdf.

    In this paper we provide an error analysis of a subgrid scale eddy viscosity method using discontinuous polynomial approximations, for the numerical solution of the incompressible Navier-Stokes equations. Optimal continuous in time error estimates of the velocity are derived. The analysis is completed with some error estimates for two fully discrete schemes, that are first and second order in time respectively.

31. B. Riviere. "Analysis of a discontinuous finite element method for the coupled Stokes and Darcy problems", Journal of Scientific Computing, 22 no 1 p. 479-500, 2005; pdf .

    The coupled Stokes and Darcy flows problem is solved by the locally conservative discontinuous Galerkin method. Optimal error estimates for the fluid velocity and pressure are derived.

32. B. Riviere and I. Yotov. "Locally conservative coupling of Stokes and Darcy flows", SIAM Journal on Numerical Analysis, 42 no 5, p. 1959-1977, 2005: pdf.

    A locally conservative numerical method for solving the coupled Stokes and Darcy flows problem is formulated and analyzed. The approach employs the mixed finite element method for the Darcy region and the discontinuous Galerkin method for the Stokes region. A discrete inf-sup condition and optimal error estimates are derived.

33. P. Bastian and B. Riviere. "Superconvergence and H(div) Projection for Discontinuous Galerkin Methods", International Journal for Numerical Methods in Fluids, Volume 42 pp. 1043--1057, 2003.

    We introduce and analyze a projection of the discontinuous Galerkin (DG) velocity approximations that preserve the local mass conservation property. The projected velocities have the additional property of continuous normal component. Both theoretical and numerical convergence rates are obtained which show that the accuracy of the DG velocity field is maintained. Superconvergence properties of the DG methods are shown. Finally, numerical simulations of complicated flow and transport problem illustrate the benefits of the projection.

34. V. Girault, B. Riviere and M.F. Wheeler. "A Discontinuous Galerkin Method with Non-Overlapping Domain Decomposition for the Stokes and Navier-Stokes Problems", Mathematics of Computation, 74, p. 53-84, 2005: postscript , pdf .

    A family of discontinuous Galerkin finite element methods is formulated and analyzed for Stokes and Navier-Stokes problems. An inf-sup condition is established as well as optimal energy estimates for the velocity and L2 estimates for the pressure. In addition, it is shown that the method can treat a finite number of non-overlapping domains with non-matching grids at interfaces.

35. B. Riviere and M.F. Wheeler. "A Posteriori Error Estimates and Mesh Adaptation Strategy for Discontinuous Galerkin Methods Applied to Diffusion Problems" Computers & Mathematics with Applications , Volume 46, Number 1 pp. 141--163, 2003: postscript, pdf.

    A posteriori error estimates for locally mass conservative methods for subsurface flow are presented. These methods are based on discontinuous approximation spaces and referred as Discontinuous Galerkin methods. In the case where penalty terms are added to the bilinear form, one obtain the Non-symmetric Interior Penalty Galerkin methods. In a previous work, we proved optimal rates of convergence of the methods applied to elliptic problems. Here, h adaptivity is investigated for flow problems in 2D. We derive global explicit estimators of the error in the L2 norm and we numerically investigate an implicit indicator of the error in the energy norm. Model problems with discontinuous coefficients are presented.

36. B. Riviere, S. Shaw, M.F. Wheeler and J.R. Whiteman. "Discontinuous Galerkin finite element methods for linear elasticity and quasistatic linear viscoelasticity", Numerische Mathematik, Volume 95, Number 2 pp. 347--376, 2003 postscript , pdf

    We consider a finite element in space, and quadrature in time discretization of a compressible linear quasistatic viscoelasticity problem. The spatial discretization uses a discontinuous Galerkin finite element method based on polynomials of degree r - termed DG(r)-and the time discretization uses a trapezoidal-rectangle rule approximation to the Volterra (history) integral. Both semi- and fully-discrete a priori error estimates are derived without recourse to Gronwall's inequality, and therefore the error bounds do not show exponential growth in time. Moreover, the convergence rates are optimal in both h and r providing that the finite element space contains a globally continuous interpolant to the exact solution. When this is not the case, the convergence rate is suboptimal in r but remains optimal in h. We also consider a reduction of the problem to standard linear elasticity where similarly optimal a priori error estimates are derived for the DG(r) approximation.

37. E.W. Jenkins; B. Riviere and M.F. Wheeler. "A Priori Error Estimates for Mixed Finite Element Approximations of the Acoustic Wave Equation". SIAM Journal on Numerical Analysis , pdf , Volume 40, Number 5 pp. 1698--1715, 2002.

    In this paper we derive optimal a priori L_infty(L2) error estimates for mixed finite element displacement formulations of the acoustic wave equation. The computational complexity of this approach is equivalent to the traditional mixed finite element formulations of the second order hyperbolic equations in which the primary unknowns are pressure and the gradient of pressure. However, the displacement formulations with the physical variables of interest, displacement and pressure, requires less regularity on the displacement.

38. B. Riviere and M.F. Wheeler. "Coupling Locally Conservative Methods for Single Phase Flow", Computational Geosciences, Volume 6 number 3 pp.269--284 2002: postscript , pdf ,

    This works presents the coupling of two locally conservative methods for elliptic problems: namely, the discontinuous Galerkin method and the mixed finite element method. The couplings can be defined with or without interface Lagrange multipliers. The formulations are shown to be equivalent. Optimal error estimates are given; penalty terms may or may not be included. In addition, the analysis for non-conforming grids is also discussed.

39. B. Riviere and M.F. Wheeler. "Discontinuous Galerkin Methods for Flow and Transport Problems in Porous Media", Communications in Numerical Methods in Engineering , 18 p. 63--68 (2002). postscript , pdf .

    This work presents a new scheme based on discontinuous approximation spaces for solving the miscible displacement problem in porous media. Numerical comparisons are made between this scheme and the well known mixed finite element and higher order Godunov methods. The simulations clearly show the advantages of the discontinuous Galerkin methods for stable or unstable flows.

40. B. Riviere; M.F. Wheeler and V. Girault. "A Priori Error Estimates for Finite Element Methods based on Discontinuous Approximation Spaces for Elliptic Problems" SIAM Journal on Numerical Analysis, volume 39 number 3 (2001) pp 902-931. postscript, pdf.

    We analyze three discontinuous Galerkin approximations for solving elliptic problems in two or three dimensions. In each one, the basic bilinear form is nonsymmetric: the first one has a penalty term on edges, the second has one constraint per edge, and the third is totally unconstrained. For each of them we prove hp error estimates in the H1 norm, optimal with respect to h, the mesh size, and nearly optimal with respect to p, the degree of polynomial approximation. We establish these results for general elements in two and three dimensions. For the unconstrained method, we establish a new approximation result valid on simplicial elements. L2 estimates are also derived for the three methods.

41. B. Riviere; M.F. Wheeler and K. Banas. " Part II. Discontinuous Galerkin Method Applied to Single Phase Flow in Porous Media", Computational Geosciences, volume 4 number 4, pp 337-341 (2000). postscript, pdf .

    Discontinuous Galerkin numerical simulations of single phase flow problems are described in this paper. The simulations show the advantages of using discontinuous approximation spaces. hp convergence results are obtained for smooth solutions. Unstructured meshes and unsmooth solutions are also considered.

42. B. Riviere; M.F. Wheeler and V. Girault. "Improved Energy Estimates for Interior Penalty, Constrained and Discontinuous Galerkin Methods for Elliptic Problems. Part I". Computational Geosciences , volume 8, pp 337-360, April 1999. postscript, pdf .

    Three Galerkin methods using discontinuous approximation spaces are introduced to solve elliptic problems. The underlying bilinear form for all three methods is the same and is nonsymmetric. In one case, a penalty is added to the form and in another, a constraint on jumps on each face of the triangulation. All three methods are locally conservative and the third one is not restricted. Optimal a priori error estimates are derived for all three procedures.

43. B. Riviere. "Analysis of a multi-numerics/multi-physics problem". Proceedings of ENUMATH 2003, to appear. postscript.
44. S. Sun, B. Riviere and M.F. Wheeler. "A combined mixed finite element and discontinuous Galerkin method for miscible displacement problem in porous media". Recent Progress in Computational and Applied PDEs, Proceedings in Recent Progress in Computational and Applied PDEs, to appear.
45. B. Riviere and M.F. Wheeler. "Discontinuous Finite Element Methods for Acoustic and Elastic Wave Problems" ICM2002-Beijing Satellite Conference on Scientific Computing, Contemporary Mathematics 329, AMS pp. 271--282, 2003. postscript.

    In this paper we formulate and analyze a family of discontinuous spatial discretizations for approximating the solution to elastic and acoustic wave problems. These schemes have the property of being able to treat highly varying material properties as well as satisfying the momentum equation locally. Here, a priori error estimates in energy and L2 are derived.

46. B. Riviere and M.F. Wheeler. "Non Conforming Methods for Transport with Nonlinear Reaction". In Fluid Flow and Transport in Porous Media: Mathematical and Numerical Treatment, Chen Z., Ewing R.E. (eds), Contemporary Mathematics, volume 295, pp 421-432, 2002. postscript, pdf.

    The transport equation is solved by a discontinuous Galerkin method, that is locally conservative and that allows for non-conforming meshes. The convective fluxes are upwinded. hp error estimates are derived in L^{\infty}(L^2) and L2(H1) for the continuous in time scheme. A class of fully discrete schemes is presented and analyzed.

47. B. Riviere and M.F. Wheeler. "Locally conservative algorithms for flow" MAFELAP 1999 Proceedings, pp 29-46, 2000.
48. B. Riviere and M.F. Wheeler. "A discontinuous Galerkin methods applied to nonlinear parabolic equations". Discontinuous Galerkin Methods: Theory, Computation and Applications, volume 11, pp 231-244, July 1999.
49. G.Baker, J. Gunnels, G. Morrow, B. Riviere, R. Van De Geijn. "PLAPACK: High performance through high level abstraction". Proceedings of the 1998 International Conference on Parallel Processing, 1998.

Other selected publications

1. B. Riviere. "Numerical study of a discontinuous Galerkin method for incompressible two-phase flow", ECCOMAS Proceedings, 2004.

   This paper presents a high-order numerical method for solving the pressure-saturation formulation of the two-phase flow problem. The saturation and pressure of the wetting phase are approximated by totally discontinuous polynomials of different order. The robustness of the method is shown for homogeneous and heterogeneous porous media.

2. M.F. Wheeler, M. Peszynska and B. Riviere. "Computational science issues in modeling oil and gas production". Proceedings of the 8th European Conference on the Mathematics of Oil Recovery- ECMOR VIII, publisher EAGE, to appear.
3. B. Riviere. "The DGIMPES Model in IPARS: Discontinuous Galerkin for Two-Phase Flow Integrated in a Reservoir Simulator Framework". TICAM Report 02-29, Austin 2002.
4. M. Guillot, B. Riviere and M.F. Wheeler. "An Implementation of a Discontinuous Galerkin Discretization of the Mass Conservation Equations in CEQUAL-ICM". TICAM Report 02-13, Austin 2002.
5. B. Riviere; E. Jenkins. "In Pursuit of Better Models and Simulations, Oil Industry Looks to the Math Sciences". SIAM News, January/February 2002.
6. B. Riviere and M.F. Wheeler. "Miscible displacement in porous media". In Computational Methods in Water Resources, Developments in Water Science, Hassanizadeh S.M., Schotting R.J., Gray W.G., Pinder G.F. (eds), Elsevier, pp 907-914, 2002. postscript, pdf.
7. M. Guillot, B. Riviere and M.F. Wheeler. "Discontinuous Galerkin methods for mass conservation equations for environmental modeling". In Computational Methods in Water Resources, Developments in Water Science, Hassanizadeh S.M., Schotting R.J., Gray W.G., Pinder G.F. (eds), Elsevier, pp 939-946, 2002. postscript, pdf.
8. E. Jenkins; B. Riviere. "Geoscientists Meet in Colorado to Explore Increasingly Complex, Multidisciplinary Problems", SIAM News, 24 (9) November 2001.
9. B. Riviere, M.F. Wheeler and E. Jenkins. "Locally conservative algorithms for flow". Proceedings of the Department of Defence User's Group Conference, 2001.
10. B. Riviere and M.F. Wheeler. "Discontinuous Finite Element Methods for Acoustic and Elastic Wave Problems. Part I: Semidiscrete Error Estimates", TICAM Report 01-02, Austin, 2001.
11. C. Dawson; B. Riviere and M.F. Wheeler. "Discontinuous Galerkin Methods for Flow and Reactive Transport". Proceedings of the Department of Defense. User's Group Conference. , June 4-7, 2000.
12. B. Riviere and M.F. Wheeler. "Optimal Error Estimates for Discontinuous Galerkin Methods Applied to Linear Elasticity Problems", TICAM Report 00-30, Austin, 2000.
13. B. Riviere; K. Banas and M.F. Wheeler. "hp 3D Flow Simulations with Discontinuous Galerkin Methods", TICAM Report 00-29, Austin, 2000.
14. B. Riviere and M.F. Wheeler. "A Posteriori Error Estimates and Mesh Adaptation Strategy for Discontinuous Galerkin Methods Applied to Diffusion Problems", TICAM Report 00-10, Austin, 2000.
15. B. Riviere; M.F. Wheeler and C. Baumann. "Part II. Discontinuous Galerkin Method Applied to a Single Phase Flow in Porous Media", TICAM Report 99-10, Austin, 1999.