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：通过一种有限元方法，我们数值求解了复杂区域上的抛物型偏微分方程，避免了网格生成过程，使用规则背景网格，而非精确贴合区域及其真实边界。我们的技术遵循phi-FEM范式，该范式假设区域由水平集函数给出。本文证明了时间隐式欧拉离散下l2(H1)和linf(L2)范数的先验误差估计。我们通过数值算例展示了phi-FEM的性能，该方法兼具最优收敛精度、易于实现和快速性等优势。