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的广泛应用中是一种具有高度实用价值的工具。研究人员将研究两种新的、有前途的技术，以高精度和更有效的算法对一系列实际感兴趣的问题进行这些模拟。它们包括在航空航天和力学(热流、不可压缩流体流动、亚音速和超音速流动)以及土木工程和机械工程(地震波传播和固体结构的弹性动力学)中的许多应用。研究人员建议继续开发一种最近发现的伴随恢复方法，该方法可以将线性和非线性泛函的Galerkin方法的近似计算成本降低几个数量级。结合自适应技术，处理解的不同程度的正则性，并将这种方法推广到非线性泛函的计算，如微分算子的特征值，将使该方法具有很大的实用价值。研究人员还将开发具有哈密顿结构的偏微分方程的新的离散化技术。它们将超收敛的间断Galerkin空间离散与辛时间推进格式相结合，得到了能量不漂移的方法。这种方法在许多应用中都很重要，包括地震波传播和弹性动力学。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.