Superconvergence of triangular quadratic finite element with interpolated coefficients for semilinear parabolic equation (Q879569)

The authors consider the numerical solution of the problem \[ u_t+Au+f(u)=g\text{ in }\Omega \times (0, T),\;u=0\text{ on }\partial \Omega \times (0, T),\;u(x,0)=u_0\text{ in }\Omega,\tag{1} \] where \(\Omega\) is a bounded complex polygonal domain in \(\mathbb{R}^2\), \(A\) is an elliptic operator, \(g \in L^2(\Omega)\), \(f'(u)\geq 0\) and \(f''(u) \in C(\mathbb{R})\). For this purpose they build the spatially semidiscrete approximation of (1) using the triangular quadratic finite element method with interpolated coefficient (ICFEM). Based on a class of orthogonal expansion in triangles, it is shown that ICFEM has superconvergence \(O(h^4\ln h)\) at each vertex and the side midpoints of all triangles.