Partial differential equations (PDEs) provide the principle mathematical models of physical phenomena that undergo continuous spatial and/or temporal variation, and the approximation of their solutions is of fundamental importance in a broad spectrum of scientific and engineering applications. Finite element methods (FEM) for solving PDEs are favored in the scientific community because of their flexibility in representing materials with complex geometries and spatially-varying material properties, and their ability to resolve local features of the solution, such as sharp transitions (e.g. shocks, layers) and singularities (unbounded derivatives). FEM works by partitioning the region of interest into a mesh consisting of smaller computational cells, and approximating the solution of the PDE in terms of â€œsimpleâ€ functions defined on these cells. This project aims to increase the flexibility of FEM by allowing for significantly more general cell shapes and function types, to more efficiently model complex materials. Publicly-available software will be produced, together with supporting mathematical theory and numerical experiments illustrating its practical performance on problems exhibiting challenging and realistic features.
The PI and collaborators will develop theory and practical algorithms for computing with finite elements defined on meshes consisting of rather general curvilinear polygons. Among the issues that will be addressed are: (i) interpolation/approximation theory for the resulting finite element spaces, which contain special locally-harmonic functions in addition to local polynomials; (ii) basis selection and efficient and accurate quadrature rules for assembling the finite element linear system; (iii) efficient and reliable a posteriori error estimators, and self-adaptive refinement techniques based on them; (iv) development of freely-available software, hosted on a public repository, that includes example problems. The types of problems that motivate this work are those involving PDE models of complex materials that may have multiple complicated interfaces between material types. Numerical examples supplied in the articles will highlight the performance of the proposed method on such problems, making relevant comparisons with competing approaches where feasible.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.