We present and analyze three distinct semi-discrete schemes for solving nonlocal geometric flows incorporating perimeter terms. These schemes are based on the finite difference method, the finite element method, and the finite element method with a specific tangential motion. We offer rigorous proofs of quadratic convergence under $H^1$-norm for the first scheme and linear convergence under $H^1$-norm for the latter two schemes. All error estimates rely on the observation that the error of the nonlocal term can be controlled by the error of the local term. Furthermore, we explore the relationship between the convergence under $L^\infty$-norm and manifold distance. Extensive numerical experiments are conducted to verify the convergence analysis, and demonstrate the accuracy of our schemes under various norms for different types of nonlocal flows.

翻译：本文针对包含周长项的非局部几何流，提出并分析了三种不同的半离散格式。这些格式分别基于有限差分法、有限元法以及带有特定切向运动的有限元法。我们严格证明了第一种格式在$H^1$范数下具有二次收敛性，后两种格式在$H^1$范数下具有线性收敛性。所有误差估计均依赖于以下观察：非局部项的误差可由局部项的误差控制。此外，我们还探讨了$L^\infty$范数收敛性与流形距离之间的关系。通过大量数值实验验证了收敛性分析，并展示了所提格式在不同范数下针对不同类型非局部流的精度。