turkmath.org

The geometric discretizations of nonlinear elasticity and fluid mechanics have mostly performed by forming locally defined series of piecewise polynomials. Many problems of continuum mechanics where solutions are smoothly varying in space call for a spectral numerical treatment as it converges exponentially by leveraging fast implementations of transforms such as the Fast Fourier Transform. Even though, spectral methods are theoretically formulated in early 80's however, due to lack of rigorous mathematical formulation, it is not much preferred for complex geometries which is commonly encountered in many engineering applications such as fluid mechanics and nonlinear elasticity. Preserving the underlying geometric, topological, and algebraic structures partial differential equations in discrete form has proven to be an effective guiding principle for numerical methods in such mechanics problems. Although the structure-preserving methods have traditionally used spaces of piecewise polynomial basis functions for differential forms, recently a structure-preserving numerical tools with spectral accuracy on logically rectangular grids over periodic or bounded domains, so called Spectral Exterior Calculus were presented. In ongoing work, its extension to nonlinear continuum mechanics problems is discussed.