We consider a multiscale elliptic eigenvalue problem that depends on n separable microscopic scales. Using multiscale homogenization, we derive the multiscale homogenized eigenvalue problem whose solution contains all the possible eigenvalues and eigenfunctions of the homogenized eigenvalue problem. We develop the sparse tensor product finite element (FE) method for solving this multiscale homogenized problem, thus bypassing the expensive task of forming the homogenized equation. Although the multiscale homogenized eigenvalue problem is posed in a high dimensional tensorized domain, the sparse tensor product FEs achieve a prescribed level of accuracy requiring an essentially optimal number of degrees of freedom which differs from the optimal one by only a possible logarithmic multiplying factor, given that the solution to this problem is sufficiently regular. We show that the regularity requirement is achieved under the Lipschitz condition on the multiscale coefficient. Numerical examples on two- and three-scale eigenvalue problems support the theoretical error estimates.

我们考虑一个依赖于n的多尺度椭圆特征值问题可分离的微观尺度。使用多尺度均质化，我们推导出多尺度均质化特征值问题，其解包含均质化特征值问题的所有可能的特征值和特征函数。我们开发了稀疏张量积有限元 (FE) 方法来解决这个多尺度均质化问题，从而绕过了形成均质化方程的昂贵任务。虽然多尺度均质化特征值问题是在高维张量域中提出的，但稀疏张量积 FE 达到了规定的精度水平，需要一个本质上最佳的自由度数，该自由度与最佳自由度仅相差一个可能的对数乘法因子，给定这个问题的解决方案是足够有规律的。我们表明在多尺度系数的 Lipschitz 条件下实现了规律性要求。两尺度和三尺度特征值问题的数值例子支持理论误差估计。