We present a Lagrange-Galerkin scheme free from numerical quadrature for convection-diffusion problems. Since the scheme can be implemented exactly as it is, theoretical stability result is assured. While conventional Lagrange-Galerkin schemes may encounter the instability caused by numerical quadrature error, the present scheme is genuinely stable. For the P kelement we prove error estimates of O(∆t + h 2 + h k+1 ) in ℓ ∞ (L 2 )-norm and of O(∆t + h 2 + h k ) in ℓ ∞ (H 1 )-norm. Numerical results reflect these estimates.Date: April 18, 2015. 2000 Mathematics Subject Classification. 65M12 and 65M25 and 65M60 and 76M10 . Key words and phrases.Lagrange-Galerkin scheme and Finite element method and Convection-diffusion problems and Exact integration . 1Stability and error analysis of LG schemes has been done in [1,3,4,6,9,10,11,12,13,14,16]; see also the bibliography therein. Pironneau [11] analyzed convection-diffusion problems and the Navier-Stokes equations to obtain suboptimal convergence results. Optimal convergence results were obtained by Douglas-Russell [6] for convection-diffusion problems and by Süli [16] for the Navier-Stokes equations. Optimal convergence results of second order in time were obtained by Boukir et al. [4] for the Navier-Stokes equations in multi-step method and by for convection-diffusion problems in single-step method. All these theoretical results are derived under the condition that the integration of the composite function term is computed exactly. Since, in real problems, it is difficult to get the exact integration value, numerical quadrature is usually employed. It is, however, reported that instability may occur caused by numerical quadrature error in [9,17,18]. That is, the theoretical stability results may collapse by the introduction of numerical quadrature.Several methods have been studied to avoid the instability. The map of a particle from a time to the previous time along the trajectory, which is nothing but to solve a system of ordinary differential equations (ODEs), is simplified in [3,9,13]. Morton-Priestley-Suli [9] solved the ODEs only at the centroids of the elements, and Priestley [13] did only at the vertices of the elements. The map of the other points is approximated by linear interpolation of those values. It becomes possible to perform the exact integration of the composite function term with the simplified map. Bermejo-Saavedra [3] used the same simplified map as [13] to employ a numerical quadrature of high accuracy to the composite function term. Tanaka-Suzuki-Tabata [19] approximated the map by a locally linearized velocity and the backward Euler approximation for the solution of the ODEs in P 1 -element. The approximate map makes possible the exact integration of the composite function term with the map. Pironneau-Tabata [12] used mass lumping in P 1 -element to develop a scheme free from quadrature for convection-diffusion problems.In this paper we prove the stability and convergence for the scheme with the same approximate map as [19] in P k -element fo...
We consider a pressure-stabilized Lagrange-Galerkin scheme for the transient Oseen problem with small viscosity. In the scheme we use the equal-order approximation of order k for both the velocity and pressure, and add a symmetric pressure stabilization term. We show an error estimate for the velocity with a constant independent of the viscosity if the exact solution is sufficiently smooth. Numerical examples show high accuracy of the scheme for problems with small viscosity.
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