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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 + t 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 t is large enough the convergence is of the form O( h m+1 t + h 2 + t q ); (3) as in case (2) but with t small, then the order of convergence is now O(h m + h 2 + t 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.

Abstract. We introduce a second order in time modified Lagrange-Galerkin (MLG) method for the time dependent incompressible Navier-Stokes equations. The main ingredient of the new method is the scheme proposed to calculate in a more efficient manner the Galerkin projection of the functions transported along the characteristic curves of the transport operator. We present error estimates for velocity and pressure in the framework of mixed finite elements when either the mini-element or the P 2/P 1 Taylor-Hood element are used.

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