The proposed research builds general theory, fast new algorithms and efficient open-source implementations for three computational methods in elliptic interface problems: 1. Locally-corrected spectral methods solve general elliptic systems with complex interfaces by the following new techniques. Distributional potential theory converts stable first-order formulations into bounded invertible operator equations. Ewald summation splits elliptic fundamental solutions into singular local corrections and smooth global terms. Off-diagonal low-rank structure couples with singular quadrature and graded meshes to enable fast direct solvers. 2. Geometric nonuniform fast Fourier transforms evaluate sparse sets of Fourier coefficients for distributions in elliptic solvers, invert boundary integral operators, and address a variety of computational problems. 3. Efficient contouring and distancing algorithms convert between convenient explicit and robust implicit interface representations. They employ fast computational geometry, global topology resolution, and high-order local piecewise-polynomial parametrizations. Solvers for elliptic systems with complex 3D piecewise-smooth interfaces convert explicit CAD formats to implicit representations for robust topological analysis.
A vast array of technological processes, from global warming to cancer therapy, from shape optimization to crack propagation, are modeled by elliptic systems of partial differential equations with complex material interfaces. Efficient general computational techniques for elliptic systems are essential in designing better models. Many existing codes for specific systems and interfaces use computational resources efficiently, but require excessive reprogramming to investigate new models and thus hamper scientific inquiry. The proposed research will develop, analyze, implement and disseminate a collection of robust, efficient and accurate new computational methods for the solution of general elliptic systems with complex interfaces. This interdisciplinary enterprise involves researchers and students in mathematics, physics, engineering and computer science, and benefits a wide range of scientific endeavors.
In industrial problems ranging from crystal growth to airfoil design, physical phenomena are modeled by elliptic systems of partial differential equations which can be solved only by approximate computational methods. Often these equations, while themselves stationary in time, are posed inside, and on interfaces between, complex moving domains in three-dimensional space. The resulting computational demands challenge existing methods to provide the accurate solutions required to predict and analyze industrial processes. We have advanced computational methods for elliptic systems of partial differential equations in three new directions: (1) locally-corrected spectral methods solve such systems by separating global and local influences and employing appropriately optimized methods for each, (2) least-squares fitting similarly localizes the challenging issue of matching boundary conditions for these systems by combinations of global functions, and (3) nonuniform fast Fourier transforms for the global analysis of these systems have been extended for the first time to treat data with more complex geometric structure than simple points. Open-source software and journal articles for methods (1) and (3) is in preparation, and will be released in early 2014. Theoretical analysis of advance (2) has been published, and preliminary experiments suggest that applications to the emerging field of isogeometric analysis will be fruitful.
