Superconvergent Approximations by Galerkin Methods for Partial Differential Equations

物理现象的计算机模拟在工程和物理学的广泛应用中是一种非常有价值的实用工具。研究人员将研究两种新的，有前途的技术进行这些模拟与高度准确和更有效的算法，为广泛的实际利益的问题。它们包括航空航天和机械的许多应用（热流，不可压缩流体流，亚音速和超音速流）以及土木和机械工程（地震波传播和固体结构的弹性动力学）。研究人员建议继续开发最近发现的伴随-恢复方法，它可以减少计算成本的近似的线性和非线性泛函的Galerkin方法的几个数量级。自适应技术的结合，以处理不同程度的正则性的解决方案，并扩展这种方法的非线性泛函的计算，如微分算子的特征值，将使该方法具有很大的实用价值。研究人员还将开发新的离散技术的偏微分方程的哈密顿结构。他们结合联合收割机超收敛间断Galerkin空间离散辛时间推进计划，并导致在方法的能量不漂移。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The computer simulation of physical phenomena is a highly valued tool of practical interest in a wide variety of applications in Engineering and Physics. The investigator will study two new, promising techniques of carrying out these simulations with highly accurate and more efficient algorithms for a wide range of problems of practical interest. They include many applications to Aerospace and Mechanics (heat flow, incompressible fluid flow, subsonic and supersonic flow) as well as to Civil and Mechanical Engineering (seismic wave propagation and elastodynamics of solid structures).The investigator proposes to continue to develop a recently uncovered adjoint-recovery method which can reduce the cost of computing approximations of linear and nonlinear functionals by Galerkin methods by several orders of magnitude. The incorporation of adaptivity techniques, to deal with the varying degree of regularity of the solution, and the extension of this approach to the computation of nonlinear functionals, like the eigenvalues of a differential operator, will render the method of great practical value. The investigator will also develop new discretization techniques for partial differential equations with Hamiltonian structure. They combine superconvergent discontinuous Galerkin space discretizations with symplectic time-marching schemes and result in methods with an energy which does not drift. Such methods are important in many applications including seismic wave propagation and elastodynamics.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.