After introducing the methods for computing covariant differential operators in the preceding chapters, we now incorporate moving frames into the numerical solution of partial differential equations (PDEs). We are particularly interested in solving PDEs defined on complex domains. By a complex domain, we mean a non-Euclidean domain, which typically requires a curved coordinate system for reference. Depending on the number of basis vectors, this domain may be a two-dimensional (2D) curved surface or a three-dimensional (3D) curved volume. Additionally, the medium may be anisotropic—meaning that it exhibits direction-dependent properties—referred to as an anisotropic medium, where wave propagation speeds differ along specific directions. See Fig. 5.1 for illustrations.