Finite difference and finite element methods exhibit first order convergence when applied to static interface problems where the grid and interface are not aligned. Although modified and unstructured grid methods would address the issue of misalignment for finite elements, application to large models of stratified media, such as those encountered in exploration geophysics, may require not only manual mesh manipulation but also more degrees of freedom than are ultimately necessary to resolve the solution. Instead using fitted or otherwise modified grids, this thesis details an improvement to an existing upscaling method that incorporates fine-scale variations of material properties by composing standard piecewise linear basis functions with a specific type of harmonic map. This technique requires that the problem domain be discretized using two meshes: one fine mesh where the harmonic map is computed to resolve fine-scale structures, and a coarse mesh where the solution to the problem is approximated. The implementation of this method in the literature restricts these composite basis functions to triangular elements in 2D leading to a non-conforming finite element method and suboptimal convergence. However, the support of these basis functions in harmonic coordinates is triangular. I present a mesh-mesh intersection algorithm that exploits this alternative representation to determine the true support of the composite basis functions in terms of the fine mesh. The result is a conforming, high-resolution finite element basis that is associated with the original coarse mesh nodes. Leveraging this fine scale information, I develop a new finite element matrix assembly algorithm. Knowing the shape of the basis support leads naturally to an integration method for computing the finite element matrix entries that is exact up to the accuracy of the harmonic map approximation. This new conforming method is shown to improve the accuracy of solutions to elliptic PDE with discontinuous coefficients on coarse, regular grids.