多尺度势薛定谔方程算子压缩多尺度有限元法的收敛性分析
Convergence analysis of an operator-compressed multiscale finite element method for Schrödinger equations with multiscale potentials
摘要：
本文分析了半经典区域中具有一般多尺度势的薛定谔方程的算子压缩多尺度有限元法（OC MsFEM）的收敛性.在OC MsFEM中，通过求解约束能量最小化来构造多尺度基函数。在对网格步长$H$的一个温和的假设下，我们证明了多尺度基函数的指数衰减性，从而可以构造局部化的多尺度基函数，如果过采样尺寸为$m = O(\log(1/H))$，则可以达到与全局基函数相同的精度.我们证明了OC MsFEM在能量范数下的一阶收敛性和在$L^2$范数下的二阶收敛性，并且当解具有足够高的正则性时，可以获得超收敛速度.通过分析解的正则性，我们还得到了误差估计对薛定谔方程小参数的依赖性。我们发现，OC MsFEM优于有限元法（FEM），由于超收敛行为的高正则性的解决方案和弱依赖于小参数的低正则性解决方案的存在下的多尺度潜力。最后，我们提出的数值结果证明的OC MsFEM的准确性和鲁棒性。
英文摘要：
In this paper, we analyze the convergence of the operator-compressed multiscale finite element method (OC MsFEM) for Schrödinger equations with general multiscale potentials in the semiclassical regime. In the OC MsFEM the multiscale basis functions are constructed by solving a constrained energy minimization. Under a mild assumption on the mesh size $H$, we prove the exponential decay of the multiscale basis functions so that localized multiscale basis functions can be constructed, which achieve the same accuracy as the global ones if the oversampling size $m = O(\log(1/H))$. We prove the first-order convergence in the energy norm and second-order convergence in the $L^2$ norm for the OC MsFEM and super convergence rates can be obtained if the solution possesses sufficiently high regularity. By analysing the regularity of the solution, we also derive the dependence of the error estimates on the small parameters of the Schrödinger equation. We find that the OC MsFEM outperforms the finite element method (FEM) due to the super convergence behavior for high-regularity solutions and weaker dependence on the small parameters for low-regularity solutions in the presence of the multiscale potential. Finally, we present numerical results to demonstrate the accuracy and robustness of the OC MsFEM.
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