Investigation of Auxiliary Subspace Techniques as a General Tool for A Posteriori Error Estimation

A posteriori error estimation is an essential component of high-performance finite element computations. Such estimates are used in practice not only to reliably determine when an approximate solution is accurate enough, but also to (efficiently) adaptively improve the approximation. This proposal considers auxiliary subspace error estimates, which are derived from computing an approximate error function in an auxiliary space. Such an approximate error function provides great flexibility, in principle, in how it may be used for adaptive finite, and we are here concerned estimation of and adaptivity with respect to: error in a variety of norms, higher-order derivatives in a variety norms, functional error for general classes of linear functionals, and error in eigenvalue and invariant subspace computations. The robustness of hierarchical error estimates of energy-norm error is well-established both in theory and in practice for low-order finite elements and secondorder linear elliptic boundary value problems in two dimensions. This proposal aims to significantly extend both theory and practice not only to include the various error measures mentioned above, but also higher-order elements in two and three dimensions (p- and hp-adaptivity), as well as to different types of operators and finite elements, including systems of partial differential equations. Additionally, an adaptive convergence theory would also be developed, where possible. A key component of the proposed research is the development of a basic framework in which clear guidance concerning an appropriate choice of auxiliary space for computing the approximate error function is provided by considering a few basic properties of the underlying problem and the space which was used for the approximate solution. Intellectual Merit: Most published analyses of the efficiency, reliability and convergence of adaptive methods do not adequately address issues relevant to the modeling composite materials or other similar problems in which the underlying differential operator might naturally have strong discontinuities and anisotropies. The proposed research would provide a general framework in which at least efficiency and reliability could naturally be established under very realistic assumptions— generally nothing more than what must be assumed for well-posedness and practical implementation. In addition to work done for higher-order Lagrange elements for standard second-order linear elliptic problems and lower order elements for some non-linear elliptic problems, implementation and careful analysis would be carried out on a variety of examples, including: Maxwell’s equations with N´ed´elec elements, the Stokes’ and related problems with Taylor-Hood elements, the bi-harmonic equation and related fourth-order problems with (non-conforming) Lagrange elements, and Monge-Amp`ere equations with (non-conforming) Lagrange elements. The proposer asserts that the general framework, and the knowledge and experience gained from these specific examples, will not only decrease the gap between what is seen in practice and the more pessimistic predictions of the current theory, but also aid in the design of related error estimators in other contexts. Broad Impact: Adaptive finite element methods are widely used by scientists in their numerical simulations, sometimes with little theory to back up their application for the particular problems under consideration. The composite materials example mentioned above is just one of many instances for which a more robust analysis is desirable. A clearer and more unified theory would make theoretical understanding of, and thereby a proper choice of, error estimators easier for practitioners; and make it easier to teach students some of the more up-to-date techniques in adaptive approximation. The various projects considered here would involve experts both in the United States and Europe, and progress would be regularly reported at national and international conferences. This proposal also includes funding for the participation of a graduate student, and thereby promotes training of the next generation of numerical analysts. Additionally, much of the software developed in conjunction with this proposal will be made freely available by the proposer from his website.