Micromechanical Analysis of Stress-Strain Inhomogeneities with Fourier transforms (MASSIF) is a large-scale Fortran-based differential equation solver used to study local stresses and strains in materials. Due to its prohibitive memory requirements, it is extremely difficult to port the code to GPUs with small on-device memory. In this work, we present an algorithm design that uses domain decomposition with approximate convolution, which reduces memory footprint to make the MASSIF simulation feasible on distributed GPU systems. A first-order performance model of our method estimates that compression and multi-resolution sampling strategies can enable domain computation within GPU memory constraints for 3D grids larger than those simulated by the current state-of-the-art Fortran MPI implementation. The model analysis also provides an insight into design requirements for further scalability. Lastly, we discuss the extension of our method to irregular domain decomposition and challenges to be tackled in the future.