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∞(L2)-norm and in the L∞(H1)-norm for the semidiscrete case whereas for the fully discrete case, we study the error analysis in the L∞(L2)-norm. Also, we check for the optimal results. For the CFE technique in the L∞(L2)-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.

中文翻译：

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在非凸域中将有限元方法变体应用于抛物线问题


在本文中，我们解决了一个主要困难，即在寻找问题的解决方案时域的非凸行为。域中出现不平滑的部分是挑战出现的地方，使用不同的数值方法处理该区域的方式揭示了这些技术的有效性。在本文中，我们研究了非凸多边形域中的半线性抛物线问题。对于解的近似，我们使用复合有限元 （CFE） 方法，这是有限元方法的一种分类。CFE 讨论了双尺度离散化 — 较大的网格也称为大小为 H 的粗网格和较小的网格，也称为大小为 h 的细网格。它有助于减少考虑的领域空间的维度。精细比例网格用于解决边界的非凸性，而粗略比例网格由距边界适当距离的较大网格组成。自由度取决于粗略网格。这是 CFE 优于其他方法的优点，即它简化了降低域复杂性的任务。在本文中，我们考虑了两种方法：半离散分析，其中仅执行空间离散化，以及完全离散分析，其中时间和空间离散化都使用反向欧拉和 Crank-Nicolson 方法完成。我们研究了半离散情况下 L∞（L2） 范数和 L∞（H1）-范数中的误差分析，而对于完全离散情况，我们研究了 L∞（L2）-范数中的误差分析。此外，我们还会检查最佳结果。 对于 L∞（L2） 范数中的 CFE 技术，我们推导出了即使域是非凸的，也具有最佳时间顺序和几乎最佳空间顺序的收敛。我们考虑一个 T 形域和另一个星形域来执行理论发现。此后，实施数值计算以验证理论结果。