Finite Element Methods for Two Problems for Hyperbolic Partial Differential Equations

Florida State University, Tallahassee FL

Investigators

Abstract

Finite element methods are often the method of choice for determining approximate solutions of partial differential equations (PDEs) and they are usually also the most thoroughly analyzed. However, this is not the case in the setting of nonlinear hyperbolic conservation laws, e.g., for compressible, inviscid flows and many other applications. The goal of the first project is to develop stable, efficient, and accurate finite element methodologies for this setting. It involves the use of hierarchical finite element bases (HFEBs). The principles used are similar to those that serve to define spectral viscosity methods. The goal is to maintain, at least away from discontinuities, the full accuracy of the finite element discretization while simultaneously suppressing unwanted and unstable oscillations. HFEBs have important advantages over both standard finite element bases and spectral bases. The separation of scales inherent in HFEBs (but which is totally absent from standard bases) allows for the selective application of artificial viscosity to only the high frequency components of the discrete solution. On the other hand, HFEB functions defined with respect to the finest grid levels are locally supported. This leads to the easy identification of the positions of discontinuities (something difficult to do with spectral bases) which, in turn, enables the efficient implementation of grid refinement strategies and the spacially selective application of artificial diffusion. All of these desirable features lead to the possibility of developing stable, high-accuracy methods for hyperbolic conservation laws. Analytical and computational studies will be carried for multi-dimensional hyperbolic conservation laws. Among the important algorithmic questions that need to be answered are how to best choose the frequency and spacially-dependent artificial viscosity coefficient and how to efficiently implement higher-order HFEBs. The second project deals with exact controllability problems for hyperbolic PDEs that impact, among many other applications, the stabilization of vibrating structures and the reduction of aerodynamically induced noise. Some preliminary studies have resulted in efficient finite difference algorithms for one-dimensional and geometrically simple two-dimensional wave equation problems. The algorithms are based on discretizing the wave equation and the initial and terminal conditions to produce and underdetermined linear system. This system acts as a constraint for an optimization problem which is defined in order to extract particular solutions of the controllability problem. Compared to previous approaches, the new method does not require regularization to obtain convergent approximations. The proposed work includes the extension of the new algorithmic approach to the finite element setting so that more complicated and realistic geometries can be treated. Extensions to equations with variable coefficients and to systems of hyperbolic PDEs, e.g., the equations of linear elasticity, will also be considered. The algorithmic studies will be complemented with analyses of the convergence properties of the optimization-based methods. The proposed projects are directly related to important problems in many applications so that their completion should have significant impact on how those problems are solved. For example, although huge efforts have been devoted to developing computational techniques for supersonic flows, there is still a need for developing high-accuracy methods that can be implemented for general geometries such as flows about airplanes. The first proposed project has exactly this goal, and its successful completion would directly impact the way problems are solved in the aerospace, nuclear, geophysics, and other communities. Furthermore, for the setting of controlling the unstable motions of structures or for the reduction in noise produced by engines, there are few existing algorithms, despite the fact that that setting is directly related to problems in the design of bridges, airplanes, buildings, transmission towers, etc. Thus, the successful completion of the second project, which has as its goal the development, implementation, and analysis of efficient and accurate computational methods, would have great impact on these and other applications.