  Many tasks in geometry processing are modeled as variational problems solved
numerically using the finite element method. For solid shapes, this requires a
volumetric discretization, such as a boundary conforming tetrahedral mesh.
Unfortunately, tetrahedral meshing remains an open challenge and existing
methods either struggle to conform to complex boundary surfaces or require
manual intervention to prevent failure. Rather than create a single volumetric
mesh for the entire shape, we advocate for solid geometry processing on
deconstructed domains, where a large and complex shape is composed of
overlapping solid subdomains. As each smaller and simpler part is now easier to
tetrahedralize, the question becomes how to account for overlaps during problem
modeling and how to couple solutions on each subdomain together algebraically.
We explore how and why previous coupling methods fail, and propose a method
that couples solid domains only along their boundary surfaces. We demonstrate
the superiority of this method through empirical convergence tests and
qualitative applications to solid geometry processing on a variety of popular
second-order and fourth-order partial differential equations.
