标题： 二阶抛物型方程的弥散域有限元法误差分析 显示英文标题

标题： Error analysis of the diffuse domain finite element method for second order parabolic equations

摘要： 本文分析了扩散域有限元方法(DDFE)，用以求解定义在一般不规则区域的一类二阶抛物型偏微分方程。 所提出的方法首先应用扩散域方法(DDM)和相场函数，将目标抛物型方程扩展到一个更大的矩形域上定义的类似问题，该矩形域包含原始物理域。 然后利用空间中的连续分段双线性基函数和时间上的BDF2格式对该变换后的方程进行离散化，得到一个完全离散的数值格式。 基于加权Sobolev空间，我们证明了当界面厚度参数趋于零时，DDM解收敛到原始解，并且相应的 $L^2$ 和 $H^1$ 范数下的逼近误差也得到了。 此外，在 $H^1$ 范数下还获得了完全离散的DDFE格式的最佳误差估计。 最后通过各种数值实验验证了理论结果，并展示了所提出方法的性能。 显示英文摘要

摘要： In this paper, we analyze the diffuse domain finite element method (DDFE) to solve a class of second-order parabolic partial differential equations defined in general irregular domains. The proposed method first applies the diffuse domain method (DDM) with a phase-field function to extend the target parabolic equation to a similar problem defined over a larger rectangular domain that contains the original physical domain. The transformed equation is then discretized by using the finite element method with continuous piecewise multilinear basis functions in space and the BDF2 scheme in time to produce a fully discrete numerical scheme. Based on the weighted Sobolev spaces, we prove the convergence of the DDM solution to the original solution as the interface thickness parameter goes to zero, with the corresponding approximation errors under the $L^2$ and $H^1$ norms. Furthermore, the optimal error estimate for the fully discrete DDFE scheme is also obtained under the $H^1$ norm. Various numerical experiments are finally carried out to validate the theoretical results and demonstrate the performance of the proposed method.