We consider a numerical scheme for the approximation of a system that couples the evolution of a two--dimensional hypersurface to a reaction--diffusion equation on the surface. The surfaces are assumed to be graphs and evolve according to forced mean curvature flow. The method uses continuous, piecewise linear finite elements in space and a backward Euler scheme in time. Assuming the existence of a smooth solution we prove optimal error bounds both in $L^\infty(L^2)$ and in $L^2(H^1)$. We present several numerical experiments that confirm our theoretical findings and apply the method in order to simulate diffusion induced grain boundary motion.

翻译：我们考虑一个近似系统的数字方案,即将二维超表层的演化与表面反射反射方程式相配。表面假定为图表,并根据强制平均曲线流变化。这种方法在空间使用连续的、片断线性有限元素,在时间上采用后向的欧勒法。假设存在一种顺畅的解决办法,我们证明最佳误差的界限为$L ⁇ infty(L2)美元和$L2(H1)1美元。我们提出数项实验,以证实我们的理论发现,并运用该方法模拟扩散诱发的谷物边界运动。