Some interior estimates for semidiscrete Galerkin approximations for parabolic equations

Consider a solution u of the parabolic equation \[ u t + A u = f in Ω × [ 0 , T ] , {u_t} + Au = f\quad {\text {in}}\quad \Omega \times [0,T], \] where A is a second order elliptic differential operator. Let S h {S_h} ; h small denote a family of finite element subspaces of H 1 ( Ω ) {H^1}(\Omega ) which permits approximation of a smooth function to order O ( h r ) O({h^r}) . Let Ω 0 ⊂ Ω {\Omega _0} \subset \Omega and assume that u h : [ 0 , T ] → S h {u_h}:[0,T] \to {S_h} is an approximate solution which satisfies the semidiscrete interior equation \[ ( u h , t , χ ) + A ( u h , χ ) = ( f , χ ) ∀ χ ∈ S h 0 ( Ω 0 ) = { χ ∈ S h , supp χ ⊂ Ω 0 } , ({u_{h,t}},\chi ) + A({u_h},\chi ) = (f,\chi )\quad \forall \chi \in S_h^0({\Omega _0}) = \{ \chi \in {S_h},{\text {supp}}\chi \subset {\Omega _0}\} , \] where A ( ⋅ , ⋅ ) A( \cdot , \cdot ) denotes the bilinear form on H 1 ( Ω ) {H^1}(\Omega ) associated with A. It is shown that if the finite element spaces are based on uniform partitions in a specific sense in Ω 0 {\Omega _0} , then difference quotients of u h {u_h} may be used to approximate derivatives of u in the interior of Ω 0 {\Omega _0} to order O ( h r ) O({h^r}) provided certain weak global error estimates for u h − u {u_h} - u to this order are available. This generalizes results proved for elliptic problems by Nitsche and Schatz [9) and Bramble, Nitsche and Schatz [1].