On the accuracy of the finite element method plus time relaxation,

A . If u denotes a local, spatial average of u, then u ′ = u − u is the associated fluctuation. Consider a time relaxation term added to the usual finite element method. The simplest case for the model advection equation ut + ux = f (x, t) is: We analyze the error in this and (more importantly) higher order extensions and show that the added time relaxation term not only suppresses excess energy in marginally resolved scales but also increases the accuracy of the resulting finite element approximation. I In 1973 Dupont [Du73], in a landmark result, showed that in general the usual, continuous finite element method for first order hyperbolic equations converges suboptimally by one power of the mesh width h, even for infinitely smooth solutions, periodic boundary conditions and uniform meshes (see also Hedstrom [Hed79]). For less smooth solutions, it is also well known that the usual Galerkin FEM can produce highly oscillatory approximate solutions, e.g., [C79]. Even cases (for example linear elements and cubic splines) for which optimal convergence has been proven for periodic boundary conditions on uniform meshes (e.g., Dupont [Du73], Thomée and Wendroff [TW74]), optimal convergence rates are not expected on highly nonuniform meshes. Dupont's result for smooth solutions and the "wiggles" observed in tests for less smooth solutions have motivated the development of many nonstandard Galerkin methods, stabilizations and regularizations for first order hyperbolic problems and associated convection dominated, convection-diffusion equations. Among these many variations on finite element methods, in complex applications there is a special interest in regularizations that are computational inexpensive, increase accuracy and incorporate numerical realizations of important physical processes omitted in the hyperbolic model equation. This report performs a numerical analysis of one such regularization which is motivated by work of Rosena