An Optimal Uniform a Priori Error Estimate for an Unsteady Singularly Perturbed Problem

We focus ourselves on the analysis of the solution of unsteady linear 2D singularly perturbed convection–diffusion equation. This type of equation can be considered as simplified model problem to many important problems, especially to Navier– Stokes equations. The space discretization of such a problem is a difficult task and it stimulated development of many stabilization methods (e.g. streamline upwind Petrov–Galerkin (SUPG) method, local projection stabilization methods) and layer–adapting techniques (e.g. Shishkin meshes, Bakhvalov meshes). For the overview see [9] or [8]. In order to achieve optimal diffusion–uniform error estimates we employ layer adapted meshes. On these general layer adapted meshes we assume a general space discretization covering standard conforming finite element method (FEM) or consistent stabilization methods. The resulting system of ordinary differential equations is solved by discontinuous Galerkin (DG) method. Considering the space discretization on Shishkin meshes, we will follow the theory for stationary singularly perturbed problems based on the solution decomposition, which enables us to derive a priori error estimates independent of the diffusion parameter even with respect to the norms (seminorms) of the exact solution, which can be also highly dependent on the diffusion parameter. For the details see [9]. The discontinuous Galerkin (DG) method is a very popular approach for solving ordinary differential equations arising from space discretization of parabolic problems, which is based on piecewise polynomial approximation in time. Among important advantages we should mention unconditional stability for arbitrary order, which allows us to solve stiff problems efficiently, and good smoothing property, which enables us to work with inexact or rough data. For introduction to DG time discretization see e.g. [11]. In [6] and [1] the authors study DG in time and DG and local projection stabilization method, respectively, in space on standard meshes for singularly perturbed problems. The error estimates in these papers contain norms of the exact solutions which go to infinity if diffusion parameter goes to zero. There are only few papers dealing with finite elements in space on the special meshes combined with any discretization in time. While in [7] the θ–scheme as discretization in time is used, in [5] the authors study BDF time discretization.