# zbMATH — the first resource for mathematics

Analysis of a BDF-DGFE scheme for nonlinear convection-diffusion problems. (English) Zbl 1158.65068
A numerical procedure is designed to get approximate solutions of a scalar nonlinear convection-diffusion equation on a bounded polyhedral domain $$\Omega \in \mathbb{R}^d$$ ($$d=2,3$$). It consists of a spatial discretization based on the symmetric variant of the discontinuous Galerkin finite element (DGFE) method and a semi-implicit $$k$$-step backward difference formula (BDF) for integrating the resulting system of stiff ordinary differential equations (ODEs) in time. The space semi-discretization algorithm, previously analysed by the authors, gives optimal order of convergence if certain technical requirements are satisfied. With respect to the discretization of the resulting ODEs in time, the so-called implicit-explicit methods are applied. In particular, the (linear) diffusive and stabilization terms are discretized implicitly, whereas the non linear convective term is treated by an explicit extrapolation method.
An error analysis of the procedure is carried out, obtaining a priori asymptotic error estimates in the discrete $$L^{\infty}(L^2(\Omega))$$-norm and the $$L^2(H^1(\Omega)$$-seminorm with respect to the mesh size $$h$$ and time step $$\tau$$ of the form $$\mathcal{O}(h^{p+1} + \tau^k)$$ and $$\mathcal{O}(h^{p} + \tau^k)$$, respectively, with $$k=2,3$$. These theoretical results are illustrated with several numerical examples.

##### MSC:
 65M20 Method of lines for initial value and initial-boundary value problems involving PDEs 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations 35K55 Nonlinear parabolic equations
RODAS
Full Text:
##### References:
 [1] Arnold D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19(4), 742–760 (1982) · Zbl 0482.65060 [2] Arnold D.N., Brezzi F., Cockburn B., Marini L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749–1779 (2002) · Zbl 1008.65080 [3] Ascher U.M., Ruuth S.J., Wetton B.T.R.: Implicit–explicit methods for time-dependent partial differential equations. SIAM J. Numer. Anal. 32(3), 797–823 (1995) · Zbl 0841.65081 [4] Aubin J.P.: Behavior of the error of the approximate solutions of boundary value problems for linear elliptic operators by galerkin’s and finite difference methods. Ann. Scuola Norm. Sup. Pisa 21, 588–637 (1967) · Zbl 0276.65052 [5] Babuška I., Zlámal M.: Nonconforming elements in the finite element method with penalty. SIAM J. Numer. Anal. 10, 863–875 (1973) · Zbl 0266.65071 [6] Babuška I., Baumann C.E., Oden J.T.: A discontinuous hp finite element method for diffusion problems: 1-d analysis. Comput. Math. Appl. 37, 103–122 (1999) · Zbl 0940.65076 [7] Bassi F., Crivellini A., Rebay S., Savini M.: Discontinuous Galerkin solution of the Reynolds averaged Navier-Stokes and k-{$$\omega$$} turbulence model equations. Comput. Fluids 34, 507–540 (2005) · Zbl 1138.76043 [8] Bassi F., Rebay S.: A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier–Stokes equations. J. Comput. Phys. 131, 267–279 (1997) · Zbl 0871.76040 [9] Bassi, F., Rebay, S.: A high order discontinuous Galerkin method for compressible turbulent flow. In: Cockburn, B., Karniadakis, G.E., Shu, C.W. (eds.) Discontinuous Galerkin Method: Theory Computations and Applications, Lecture Notes in Computational Science and Engineering, vol. 11. Springer, Berline, pp. 113–123 (2000) · Zbl 0991.76039 [10] Baumann C.E., Oden J.T.: A discontinuous hp finite element method for the Euler and Navier–Stokes equations. Int. J. Numer. Methods Fluids 31(1), 79–95 (1999) · Zbl 0985.76048 [11] Ciarlet P.G.: The Finite Elements Method for Elliptic Problems. North-Holland, Amsterdam (1979) [12] Cockburn B.: Discontinuous Galerkin methods for convection dominated problems. In: Barth, T.J., Deconinck, H.(eds) High-Order Methods for Computational Physics, Lecture Notes in Computational Science and Engineering, vol. 9, pp. 69–224. Springer, Berlin (1999) · Zbl 0937.76049 [13] Cockburn B., Karniadakis G.E., Shu C.W., (eds): Discontinuous Galerkin Methods. Springer, Berlin(2000) · Zbl 0989.76045 [14] Crouzeix M.: Une méthode multipas implicit-explicit pour l’approximation des équations d’évolutions paraboliques. Numer. Math. 35, 27–276 (1980) · Zbl 0419.65057 [15] Dolejší V.: On the discontinuous Galerkin method for the numerical solution of the Navier–Stokes equations. Int. J. Numer. Methods Fluids 45, 1083–1106 (2004) · Zbl 1060.76570 [16] Dolejší V.: Higher order semi-implicit discontinuous Galerkin finite element schemes for nonlinear convection–diffusion problems. In: de Castro, A.B., Gómez, D., Quintela, P., Salgado, P.(eds) Numerical Mathematics and Advanced Applications, ENUMATH 2005, pp. 432–439. Springer, Berlin (2006) · Zbl 1119.65387 [17] Dolejší V., Feistauer M.: Error estimates of the discontinuous Galerkin method for nonlinear nonstationary convection–diffusion problems. Numer. Funct. Anal. Optim. 26(25–26), 2709–2733 (2005) [18] Dolejší V., Feistauer M., Hozman J.: Analysis of semi-implicit DGFEM for nonlinear convection–diffusion problems. Comput. Methods Appl. Mech. Eng. 196, 2813–2827 (2007) · Zbl 1121.76033 [19] Dolejší, V., Feistauer, M., Kučera, V., Sobotíková, V.: An optimal L L 2)-error estimate of the discontinuous Galerkin method for a nonlinear nonstationary convection–diffusion problem. IMA J. Numer. Anal. (published online doi: 10.1093/imanum/drm023 , 2007) [20] Dolejší V., Feistauer M., Schwab C.: A finite volume discontinuous Galerkin scheme for nonlinear convection–diffusion problems. Calcolo 39, 1–40 (2002) · Zbl 1098.65095 [21] Dolejší V., Feistauer M., Sobotíková V.: A discontinuous Galerkin method for nonlinear convection–diffusion problems. Comput. Methods Appl. Mech. Eng. 194, 2709–2733 (2005) · Zbl 1093.76034 [22] Dumbser M., Munz C.D.: Building blocks for arbitrary high-order discontinuous Galerkin methods. J. Sci. Comput. 27, 215–230 (2006) · Zbl 1115.65100 [23] Feistauer M., Felcman J., Straškraba I.: Mathematical and Computational Methods for Compressible Flow. Oxford University Press, Oxford (2003) · Zbl 1028.76001 [24] Feistauer M., Hájek J., Švadlenka K.: Space–time discontinuos Galerkin method for solving nonstationary convection–diffusion-reaction problems. Appl. Math. 52(3), 197–234 (2007) · Zbl 1164.65469 [25] Feistauer M., Švadlenka K.: Discontinuous Galerkin method of lines for solving nonstationary singularly perturbed linear problems. J. Numer. Math. 12, 97–118 (2004) · Zbl 1059.65083 [26] Frank J., Hundsdorfer W., Verwer J.G.: On the stability of implicit–explicit linear multistep methods. Appl. Numer. Math. 25(6), 193–205 (1997) · Zbl 0887.65094 [27] Gear C.W.: The automatic integration of ordinary differential equations. Commun. ACM 14(3), 176–179 (1971) · Zbl 0217.21701 [28] Gear C.W.: Numerical Initial Value Problems in Ordinary Differential Equations. Prentice-Hall, Inc., Englewood Cliffs, NJ (1971) · Zbl 1145.65316 [29] Grisvard P.: Singularities in Boundary Value Problems. Springer, Berlin (1992) · Zbl 0766.35001 [30] Hairer E., Norsett S.P., Wanner G.: Solving Ordinary Differential Equations I, Nonstiff Problems. No. 8 in Springer Series in Computational Mathematics. Springer, Berlin (2000) [31] Hairer E., Wanner G.: Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems. Springer, Berlin (2002) · Zbl 0859.65067 [32] Hartmann R., Houston P.: Adaptive discontinuous Galerkin finite element methods for nonlinear hyperbolic conservation laws. SIAM J. Sci. Comput. 24, 979–1004 (2002) · Zbl 1034.65081 [33] Hartmann R., Houston P.: Symmetric interior penalty DG methods for the compressible Navier–Stokes equations. I. Method formulation. Int. J. Numer. Anal. Model. 1, 1–20 (2006) · Zbl 1129.76030 [34] Hartmann R., Houston P.: Symmetric interior penalty DG methods for the compressible Navier–Stokes equations. II. Goal-oriented a posteriori error estimation. Int. J. Numer. Anal. Model. 3, 141–162 (2006) · Zbl 1152.76429 [35] Henrici P.: Discrete Variable Methods in Ordinary Differential Equations. Wiley, New York (1962) · Zbl 0112.34901 [36] Houston P., Schwab C., Süli E.: Discontinuous hp-finite element methods for advection–diffusion problems. SIAM J. Numer. Anal. 39(6), 2133–2163 (2002) · Zbl 1015.65067 [37] Hundsdorfer W.: Partially implicit BDF2 blends for convection-dominated flows. SIAM J. Numer. Anal. 38(6), 1763–1783 (2001) · Zbl 1007.76052 [38] Klaij C.M., van der Vegt J., der Ven H.V.: Pseudo-time stepping for space–time discontinuous Galerkin discretizations of the compressible Navier–Stokes equations. J. Comput. Phys. 219(2), 622–643 (2006) · Zbl 1102.76035 [39] Klaij C.M., van der Vegt J., der Ven H.V.: Space–time discontinuous Galerkin method for the compressible Navier–Stokes equations. J. Comput. Phys. 217(2), 589–611 (2006) · Zbl 1099.76035 [40] Kufner A., John O., Fučik S.: Function Spaces. Academia, Prague (1977) [41] Lions P.L.: Mathematical Topics in Fluid Mechanics. Oxford Science Publications, USA (1996) · Zbl 0866.76002 [42] Lomtev I., Quillen C.B., Karniadakis G.E.: Spectral/hp methods for viscous compressible flows on unstructured 2d meshes. J. Comput. Phys. 144(2), 325–357 (1998) · Zbl 0929.76095 [43] Nitsche J.A.: Ein kriterium fr die quasi-optimalität des ritzschen verfahrens. Numer. Math. 11, 346–346 (1968) · Zbl 0175.45801 [44] Oden J.T., Babuška I., Baumann C.E.: A discontinuous hp finite element method for diffusion problems. J. Comput. Phys. 146, 491–519 (1998) · Zbl 0926.65109 [45] Ostermann A., Thalhammer M., Kirlinger G.: Stability of linear multistep methods and applications to nonlinear parabolic problems. Appl. Numer. Math. 48, 389–3407 (2004) · Zbl 1041.65073 [46] Rektorys K.: The Method of Discretization in Time and Partial Differential Equations. Reidel, Dodrecht (1982) · Zbl 0522.65059 [47] Rivière B., Wheeler M.F.: A discontinuous Galerkin method applied to nonlinear parabolic equations. In: Cockburn, B., Karniadakis, G.E., Schu, C.W.(eds) Discontinuous Galerkin methods Theory, computation and applications., Lect. Notes Comput. Sci. Eng., vol. 11, pp. 231–244. Springer, Berlin (2000) · Zbl 0946.65078 [48] Rivière B., Wheeler M.F., Girault V.: Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Comput. Geosci. 3(3-4), 337–360 (1999) · Zbl 0951.65108 [49] Sudirham J., van der Vegt J., van Damme R.: Space–time discontinuous Galerkin method for advection–diffusion problems on time-dependent domains. Appl. Numer. Math. 56(12), 1491–1518 (2006) · Zbl 1111.65089 [50] Varah J.M.: Stability restrictions on second order, three level finite difference scheme for parabolic equations. SIAM J. Numer. Anal. 17(2), 300–309 (1980) · Zbl 0426.65048 [51] van der Vegt J.J.W., van der Ven H.: Space–time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows. I. General formulation. J. Comput. Phys. 182(2), 546–585 (2002) · Zbl 1057.76553 [52] van der Ven H., van der Vegt J.J.W.: Space–time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows. II. Efficient flux quadrature. Comput. Methods Appl. Mech. Eng. 191, 4747–4780 (2002) · Zbl 1019.74037
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.