  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.
