Ever increasing demands on computational solution techniques necessitate development of new and improved methods. Two classes of methods, increasingly being used in simulation of physical and engineering systems, are finite element methods of the mixed type and the discontinuous Galerkin (DG) type. Building sound mathematical foundations for these methods increases their reliability, reveals avenues to improve them, and helps discover radically new methods. In this spirit, five lines of research are proposed on the following topics: (i) mixed methods (ii) discontinuous Petrov-Galerkin (DPG) schemes(iii) hybridizable discontinuous Galerkin (HDG) methods (iv) simulation of photonic membranes, and (v) complex axisymmetric simulations. The first deals with new stress elements and their implications in mixed methods for elasticity with weakly imposed stress symmetry. The second pursues a new DPG paradigm in the design of schemes where optimal test spaces are automatically computed. The third, concerns DG methods that mimic mixed methods, yet having the added advantage of flexible stabilization, and continues a line of research previously supported by the foundation. Both source problems and eigenproblems are considered. The remaining two lines of research, considers applications in need of new mathematical developments, e.g., (iv) needs good nonlinear eigensolvers and (v) needs sound treatment of singularities.
Methods for computer simulation are an indispensable tool in modern scientific research. The proposed research brings fresh mathematical ingredients that spawn novel simulation methods. These mathematical techniques have the advantage of being broadly applicable. Accordingly, several disparate application areas can be targeted, including solid mechanics, transport phenomena, fluid flow, wave propagation, triggered lightning, and nanophotonic membranes. To detail a few examples, application of the new methods to fluid flow, through industrial and academic collaborations, can potentially benefit the aircraft industry. Reliable simulation methods can inexpensively guide experimentation of next generation nanophotonic devices. Finally, human resource development is integrated into the activities through training and participation of graduate students in the research.
The combined outcomes from grants DMS-1014817 and DMS-1211635 are reported (as the first was transferred to the second). As the world demands computer simulation of increasingly complex natural and technological processes, the work of this grant focused on correctness and efficiency guarantees on the simulations that mathematics can provide, and on novel alternative simulation techniques. The research demonstrated new advantages of the `hybridization' concept in designing finite element methods of high order accuracy. In lay terms, hybridization is the process of formulating methods exploiting unknowns at the interfaces of mesh elements. One of the pursued projects showed how hybridization can be useful in eigenvalue problems: Although the correct hybridized/condensed eigenvalue problem was discovered to be nonlinear, a way to get good initial guesses was also discovered, thus making it algorithmically competitive. New properties of hybridized discontinuous Galerkin (HDG) methods were also discovered and efficient solvers for them were constructed. The disciplinary impact of hybridization extends beyond this grant, as interesting features were synergistically uncovered by other projects as well as by this grant, e.g., high number of local operations in the solution process, structured global matrices even on unstructured grids, fewer non-zeros in factored matrices, excellent compatibility with multicore architectures, flexible and transparent stabilization using an unconstrained stabilization parameter (that many authors could easily extend to nonlinear problems), sharp analytical tools that extend commuting diagram techniques from mixed methods, etc. Conferences revealed connections with other independent lines of research using interface unknowns such as CBE and WG methods. The important role of interface unknowns in designing methods seems to be well appreciated now. A highlight during this grant's period was the introduction of the discontinuous Petrov Galerkin (DPG) method. In the DPG method, a trial space is paired with an automatically computed test space that guarantees stability. The numerical studies showed that DPG methods have extraordinary stability with respect to mesh size, polynomial degrees, and some singular perturbation parameters. Theoretical studies showed that this stability arises due to the optimally paired test space, a concept previously ignored as impractical. By a judicious use of interface unknowns (or hybridization) this research made the computation of optimal test spaces practical. Ongoing research studies the application of DPG methods to fluid flow and wave propagation. An outgrowth of the grant's multidisciplinary research is a study into biological pattern formation by chemotactic phenomena. Pattern formation is important in basic biology (e.g., morphogenesis) as well as in human health (e.g., tumor growth). One publication from this grant finds a chemotaxis-based mechanism for pattern formation, distinct from the traditional one due to Turing. It was shown that homogeneous states can be destabilized by reducing an infinite dimensional eigenproblem to a parametrized finite dimensional eigenproblem. A second publication focused on positivity properties of chemotaxis models. Negative approximations of intrinsically nonnegative quantities, such as density, are erroneous, reduce confidence in simulation techniques, and generate instabilities in nonlinear iterations. However, not all nonlinear systems of partial differential equations come with a guarantee that their exact solutions are nonnegative. Their numerical discretization introduces a further layer of difficulty before one can certify that the simulated solution will be nonnegative. The paper examined and resolved these difficulties in the context of a specific nonlinear system generalizing the influential Keller-Segel chemotactic model. Five graduate students were partially supported by this grant, three of whom were from an underrepresented minority. Two of them have finished their doctoral studies and are now in the workforce as tenure-track assistant professors.
