Application of a finite element method variant in nonconvex domains to parabolic problems

Abstract

In this paper we address one of the major difficulties which is the nonconvex behavior of the domains while finding the solution of the problems. The part of the domain where the nonsmoothness appears is where the challenge arises and the way that area is handled using different numerical methods reveals the effectiveness of these techniques. Here in this article, we study the semilinear parabolic problem in nonconvex polygonal domain. For the approximation of the solution we use the Composite Finite Element (CFE) method, which is a classification of the Finite Element Method. CFE discusses the two-scale discretization — the larger mesh also known as the coarse mesh with the size $H$ and the smaller mesh, also known as the fine mesh with the size $h$. It helps in reducing the dimension of the domain space of consideration. The fine scale grid is used to resolve the nonconvexity of the boundary whereas the coarse scale grid is comprised of larger grids at an appropriate distance from the boundary. The degrees of freedom depends on the coarse grid. This is the precedence of CFE over other methods, i.e., it eases the task of reducing the domain complexity. In this article, we consider two approaches — the semi discrete analysis where only space discretization is carried out, and the fully discrete analysis where both the time and space discretization is done using both backward Euler and Crank–Nicolson method. We study the error analysis in the $L^{\infty }\left(L^2\right)$-norm and in the $L^{\infty }\left(H^1\right)$-norm for the semidiscrete case whereas for the fully discrete case, we study the error analysis in the $L^{\infty }\left(L^2\right)$-norm. Also, we check for the optimal results. For the CFE technique in the $L^{\infty }\left(L^2\right)$-norm, we derive the convergence having optimal order in time and almost optimal order in space even if the domain is nonconvex. We consider a T-shaped domain and another star shaped domain to carry out the theoretical findings. Thereafter, numerical computations are implemented to validate the theoretical results.