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.
