【摘要】 近年来,多尺度偏微分方程的数值均匀化方法得到了快速发展.本文以Rough Polyharmonic Splines （RPS）^[1]及其推广形式Generalized Rough Polyharmonic Splines （GRPS）^[2]为例,介绍了一类基于带约束能量最小基函数的数值均匀化方法的数学形式,并详细给出了基于粗细两网格,且具有拟最优计算量和收敛性的局部化基函数的数值实现方法.我们对具有多尺度系数的二维椭圆方程验证了这类方法的收敛性,此类方法在简单修改后还可用于多尺度Helmholtz方程等其他问题.更多还原

【Abstract】 Last decades have witnessed the fast development of numerical homogenization methods for multiscale PDEs.In this paper,we use Rough Polyharmonic Splines（RPS）^[1] and its generalization,namely Generalized Rough Polyharmonic Splines（GRPS） ^[2] as representative examples,to introduce the mathematical formulation and numerical implementation of a class of numerical homogenization methods with bases from constrained energy minimization.We present details of the construction of two-level mesh,local patches,and the computation of localized bases with quasi-optimal computational complexity and accuracy.The methods are numerically justified for multiscale elliptic equations with rough coefficients,and they can be applied to other problems such as multiscale Helmholtz equations with minor modifications.更多还原