Abstract
We introduce modified Lagrange–Galerkin (MLG) methods of order one and two with respect to time to integrate convection–diffusion equations. As numerical tests show, the new methods are more efficient, but maintaining the same order of convergence, than the conventional Lagrange–Galerkin (LG) methods when they are used with either P 1 or P 2 finite elements. The error analysis reveals that: (1) when the problem is diffusion dominated the convergence of the modified LG methods is of the form O(h m+1 + h 2 + Δt q), q = 1 or 2 and m being the degree of the polynomials of the finite elements; (2) when the problem is convection dominated and the time step Δt is large enough the convergence is of the form \({O(\frac{h^{m+1}}{\Delta t}+h^{2}+\Delta t^{q})}\) ; (3) as in case (2) but with Δt small, then the order of convergence is now O(h m + h 2 + Δt q); (4) when the problem is convection dominated the convergence is uniform with respect to the diffusion parameter ν (x, t), so that when ν → 0 and the forcing term is also equal to zero the error tends to that of the pure convection problem. Our error analysis shows that the conventional LG methods exhibit the same error behavior as the MLG methods but without the term h 2. Numerical experiments support these theoretical results.
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Allievi A., Bermejo R.: A generalized particle search-locate algorithm for arbitrary grids. J. Comput. Phys. 132, 157–166 (1997)
Bause M., Knabner P.: Uniform error analysis for Lagrange-Galerkin approximations of convection-dominated problems. SIAM J. Numer. Anal. 39, 1954–1984 (2002)
Bermejo R., Carpio J.: A semi-Lagrangian-Galerkin projection scheme for convection equations. IMA J. Numer. Anal. 30, 799–831 (2010)
Bernardi C.: Optimal finite-element interpolation on curved domains. SIAM J. Numer. Anal. 36, 1212–1240 (1989)
Boukir K., Maday Y., Métivet B., Razanfindrakoto E.: A high-order characteristics/finite element method for the incompressible Navier-Stokes equations. Int. J. Numer. Methods Fluids 25, 1421–1454 (1997)
Douglas J., Russell T.F.: Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with definite element of finite difference procedures. SIAM J. Numer. Anal. 19, 871–885 (1982)
Ewing R., Russell T.F.: Multistep Galerkin methods along characteristics for convection-diffusion problems. In: Vchtneveski, R., Stepleman, R.S. (eds) Advances in Computer Methods for Partial Differential Equations IV, pp. 28–36. IMACS, New Brunswick (1981)
Morton K.W., Priestley A., Süli E.: Stability of the Lagrange-Galerkin method with non-exact integration. Math. Modelling Numer. Anal. 22, 625–653 (1988)
Pironneau O.: On the transport-diffusion algorithm and its applications to the Navier-Stokes equations. Numer. Math. 38, 309–332 (1982)
Rui H., Tabata M.: A second order characteristic finite element scheme for convection-diffusion problems. Numer. Math. 92, 161–177 (2002)
Rathod H.T., Venkatesudu B., Nagaraja K.V.: Gauss-Legendre quadrature formulas over a tetrahedron. Numer. Methods Partial Differ. Equ. 22, 197–219 (2006)
Süli E.: Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations. Numer. Math. 53, 459–483 (1988)
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Bermejo, R., Saavedra, L. Modified Lagrange–Galerkin methods of first and second order in time for convection–diffusion problems. Numer. Math. 120, 601–638 (2012). https://doi.org/10.1007/s00211-011-0418-8
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DOI: https://doi.org/10.1007/s00211-011-0418-8