Composite Finite Element Approximation for Parabolic Problems in Nonconvex Polygonal Domains

Abstract

Abstract The purpose of this paper is to generalize known a priori error estimates of the composite finite element (CFE) approximations of elliptic problems in nonconvex polygonal domains to the time dependent parabolic problems. This is a new class of finite elements which was introduced by [W. Hackbusch and S. A. Sauter, Composite finite elements for the approximation of PDEs on domains with complicated micro-structures, Numer. Math. 75 1997, 4, 447–472] and subsequently modified by [M. Rech, S. A. Sauter and A. Smolianski, Two-scale composite finite element method for Dirichlet problems on complicated domains, Numer. Math. 102 2006, 4, 681–708] for the approximations of stationery problems on complicated domains. The basic idea of the CFE procedure is to work with fewer degrees of freedom by allowing finite element mesh to resolve the domain boundaries and to preserve the asymptotic order convergence on coarse-scale mesh. We analyze both semidiscrete and fully discrete CFE methods for parabolic problems in two-dimensional nonconvex polygonal domains and derive error estimates of order 𝒪 ( H s Log ^ ( H h ) s 2 ) {\mathcal{O}(H^{s}\widehat{\mathrm{Log}}{}^{\frac{s}{2}}(\frac{H}{h}))} and 𝒪 ( H 2 ⁢ s Log ^ ( H h ) s ) {\mathcal{O}(H^{2s}\widehat{\mathrm{Log}}{}^{s}(\frac{H}{h}))} in the L ∞ ⁢ ( H 1 ) {L^{\infty}(H^{1})} -norm and L ∞ ⁢ ( L 2 ) {L^{\infty}(L^{2})} -norm, respectively. Moreover, for homogeneous equations, error estimates are derived for nonsmooth initial data. Numerical results are presented to support the theoretical rates of convergence.