Biquadratic finite volume element methods based on optimal stress points for parabolic problems (Q654131)

The authors consider the following boundary value problem: \[ \frac{\partial u}{\partial t} - \nabla(a(x)\nabla u) = f(x,t),\;\;(x,t) \in \Omega \times (0, T], \] \[ u(x,t)=0, (x,t)\in \partial \Omega \times (0,T], \;\;u(x, 0)=u_0(x), \;\;x \in \Omega \] where \(\Omega\) is a rectangle, \(x = (x,y)\), \(a(x)\) is a positive real-valued function and \(f(x,t):\Omega \rightarrow \mathbb{R}\). The initial function \(u_0(x)\), the functions \(a(x), f(x,t)\) are assumed to be sufficiently smooth to ensure that the above problem has a unique solution in the appropriate Sobolev space. Semi-discrete and discrete finite volume element (FVE) schemes based on optimal stress points are constructed. Optimal order error estimates in \(H^1\) and \(L^2\) norms are derived. Also obtained are the superconvergence of numerical gradients at optimal stress points. The theoretical results are confirmed through a numerical experiment for \(a(x)=exp(x+y)\) when the exact solution of the problem is \(u(x,f)=e^{-t}\sin(x)\sin(y)\).