Thanks to a finite element method, we solve numerically parabolic partial differential equations on complex domains by avoiding the mesh generation, using a regular background mesh, not fitting the domain and its real boundary exactly. Our technique follows the phi-FEM paradigm, which supposes that the domain is given by a level-set function. In this paper, we prove a priori error estimates in l2(H1) and linf(L2) norms for an implicit Euler discretization in time. We give numerical illustrations to highlight the performances of phi-FEM, which combines optimal convergence accuracy, easy implementation process and fastness.
翻译:通过有限元方法,我们在复杂的域上数值求解抛物型偏微分方程,而不需要进行网格划分。我们使用一个规则的背景网格(不需要与实际边界精确契合)和一种称为 phi-FEM 方法的范式来实现这一点,该方法假定域由一个级集函数给出。本文证明了隐式 Euler 离散时间下的预先误差估计,其中包括了 l2(H1) 和 linf(L2) 法误差范数。我们给出了数值实验以突出 phi-FEM 方法的性能,该方法结合了最优收敛精度,易于实现的过程和快速性。