Superconvergence of a finite element method for linear integro-differential problems
The authors introduce a new way of approximating the initial condition to the semidiscrete finite element method for the following problem with memory \[ u_t=\nabla\left\{a(x,t)\nabla u+ \int_0^tb(x,t,\tau)\nabla u(x, \tau) d\tau\right\}+f(x,t),\quad (x,t)\in\Omega\times (0,T], \] \[ u(x,0) =u_0(x),\quad x\in\Omega,\quad u(x,t)=0,\quad x\in\partial \Omega\times[0,T], \] where \(\Omega\in \mathbb R^2\) is a bounded domain with boundary \(\partial \Omega\), \(a, b, f, u_0\) are bounded together with their derivatives and \(0<a_0\leq a(x,t), (x, t)\in\Omega\times[0,T].\) Several superconvergence results are obtained for the error between the approximate solution and the Ritz-Volterra projection of the exact solution. For \(k > 1,\) first order gain in \(L_p(2\leq p\leq\infty)\) norm, second order in \(W^{1,p}(2\leq p\leq\infty)\) norm and almost second order in \(W^{1,\infty}\) norm are obtained. For \(k=1,\) first order gain in \(W^{1,p}(2\leq p\leq\infty)\) norm is derived. Applying interpolated postprocessing technique to the approximate solution, one order global superconvergence between the exact solution and the interpolation of the approximate solution in the \(L_p\) and \(W^{1,p}(2\leq p\leq\infty)\) is obtained.
semidiscrete Galerkin approximation scheme
scientific article; zbMATH DE number 1485600