Error estimates of an FEM with lumping for parabolic PDEs (Q1599816)

This paper presents error estimates for the solution of a parabolic partial differential equation (PDE) employing a finite element spatial formulation and with temporal term treated specially to have a diagonal coefficient matrix. This diagonal matrix is often termed a ``lumped'' matrix. The author focusses on conforming linear trial functions on triangles, and introduces a notion of ``symmetric'' triangulation that imposes a great deal of regularity. For the partial differential equation \(\partial U/\partial t -\nabla\cdot\nabla U=f\), the spatial term is discretized following the usual finite element prescription employing integral inner products. However, for a particular mesh, the temporal term is treated with the discrete inner product \[ \langle v,w\rangle_h=(1/3)\sum_{\tau} |\tau|\sum_{i=1}^3v(P^{\tau}_i)(w(P^{\tau}_i) \] where \(\tau\) ranges over all triangles in the mesh, and \(|\tau|\) and \(P^{\tau}_i\), \(i=1,2,3\) represent the area and three vertices of the triangle \(\tau\). This discrete inner product gives rise to a diagonal coefficient matrix. The author shows that this inner product differs from the usual integral inner product by \(O(h^2)\), under fairly general assumptions, but its use introduces an extra \(O(h)\) term into error estimates. However, if the triangulation is symmetric, then \(O(h^2)\) estimates are recovered.