This paper develops and discusses a residual-based a posteriori error estimate and a space--time adaptive algorithm for solving parabolic surface partial differential equations on closed stationary surfaces. The full discretization uses the surface finite element method in space and the backward Euler method in time. The proposed error indicator bounds the error quantities globally in space from above and below, and globally in time from above and locally from below. A space--time adaptive algorithm is proposed using the derived error indicator. Numerical experiments illustrate and complement the theory.

翻译：本文针对闭合静止曲面上的抛物型曲面偏微分方程，提出并讨论了一种基于残差的后验误差估计方法及相应的时空自适应算法。该全离散化方案在空间上采用曲面有限元方法，在时间上采用后向欧拉方法。所提出的误差指示器能够从上下界两个方向全局控制空间误差量，并在时间维度上实现全局上界控制与局部下界控制。基于推导得到的误差指示器，本文提出了一种时空自适应算法。数值实验验证并补充了理论结果。