Interior maximum norm estimates for some simple finite element methods

Asymptotic error expansion and Richardson extranpolation for linear finite elements

“…If the triangulation is uniform, we obtain the following This local result also follows from the proof of Theorem 1, if one multiplies the Taylor polynomial p of u at the point z e s o by some smooth cut-off function ;(, separating f2 o from the corner points of 0f2, and observes that u eH2(f2) C 2 + e(o(.~ supp (Z))-For a uniform triangulation with rectangular triangles the above logarithm free error estimate has been proven in [5], and earlier by finite difference methods in [2]. The estimate (3.18) even holds true if the triangulation is only required to be uniform in some subdomain containing f2 o.…”

Interior maximum norm estimates for finite element methods

“…Interior maximum norm estimates, in cases where the spaces 5" are defined on uniform ("regular", "translation invariant") meshes, were given in Bramble, Nitsche and Schatz [3], Bramble and Schatz [5], Bramble and Thome'e [6], and Strang and…”

Superconvergence of the Gradient in Piecewise Linear Finite-Element Approximation to a Parabolic Problem

“…THEOREM 3.3. Let u be the solution of (1.1) and " that of (3.40) with 0=/hv, we have, for n >= 1, + k log Ilu,,(o)ll,+ Ilu.,(o)ll + (llu.,ll,+ Ilu..ll) as Before we prove this theorem, note that in the same way as for the standard Crank-Nicolson scheme, our result implies IIv(" R,U"-'/2)IIL <--C(u) h 2 log 1_+ k2 log h and ( <-_ C(u) h 2 log + k 2 We now start to prove our result. This time we may write (3.41) "-Rau" =(Jn--h--)'+'(h--Uh)bln --+(Rh-Rh)un.…”

“…Such a result may be combined with known logarithm-free maximum-norm error estimates for the elliptic problem from, e.g., Lin, Lfi, and Shen [11] (for strongly regular triangulations) or Bramble and Thome [2] (interior estimates for uniform triangulations) to derive complete O(h) maximum-norm error estimates for the parabolic problem.…”