This paper proposes novel computational multiscale methods for linear second-order elliptic partial differential equations in nondivergence-form with heterogeneous coefficients satisfying a Cordes condition. The construction follows the methodology of localized orthogonal decomposition (LOD) and provides operator-adapted coarse spaces by solving localized cell problems on a fine scale in the spirit of numerical homogenization. The degrees of freedom of the coarse spaces are related to nonconforming and mixed finite element methods for homogeneous problems. The rigorous error analysis of one exemplary approach shows that the favorable properties of the LOD methodology known from divergence-form PDEs, i.e., its applicability and accuracy beyond scale separation and periodicity, remain valid for problems in nondivergence-form.

翻译：本文针对满足Cordes条件的具有非均匀系数的线性二阶椭圆型偏微分方程（非散度形式），提出了新型计算多尺度方法。该方法沿用了局部正交分解（LOD）方法论，并通过在细尺度上求解局部化胞元问题，在数值均匀化框架下构建算子自适应粗空间。粗空间的自由度与齐次问题的非协调及混合有限元方法相关联。通过对一种典型方法的严格误差分析表明，LOD方法论在散度型偏微分方程中已知的优良特性（即超越尺度分离和周期性的适用性与精确性）在非散度形式问题中仍然成立。