In this lecture I will present an unconventional perspective on least-squares finite element methods, which connects them to compatible methods and shows that least-squares methods can enjoy the same conservation properties as their mixed Galerkin cousins.

To a casual observer, compatible (or mimetic) methods and least squares principles for PDEs couldn't be further apart. Mimetic methods inherit key conservation properties of the PDE, can be related to a naturally occurring optimization problem, and require specially selected, dispersed degrees of freedom. The conventional wisdom about least squares is that they rely on artificial energy principles, are only approximately conservative, but can work with standard C⁰ nodal (or collocated) degrees of freedom. The latter is considered to be among the chief reasons to use least squares methods.

This lecture demonstrates that exactly the opposite is true about least-squares methods. First, I will argue that nodal elements, while admissible in least squares, do not allow them to realize their full potential, should be avoided and are, perhaps, the least important reason to use least squares! Second, I will show that for an important class of problems least squares and compatible methods are close relatives that share a common ancestor, and in some circumstances compute identical answers. The price paid for gaining favorable conservation properties is that one has to give up what is arguably the least important advantage attributed to least squares methods: one can no longer use C⁰ nodal elements for all variables.

If time permits I will explore two other unconventional uses of least-squares ideas which result in numerical schemes with attractive computational properties: a least-squares mesh-tying method that passes patch tests of arbitrary orders, and a locally conservative discontinuous velocity least-squares method for incompressible flows. The material in this talk is drawn from collaborative works with M. Gunzburger (FSU), M Hyman (Tulane), L. Olson (UIUC) and J. Lai (UIUC).

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Sandia National Laboratories is a multi-program laboratory operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin company, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.

In this tutorial, we will present the hybridizable discontinuous Galerkin (HDG) methods for diffusion problems. We will describe the main idea for devising them and will explain how to implement them efficiently. We will then compare the methods with mixed methods and the continuous Galerkin methods. Finally, we will discuss the convergence properties of the methods in terms of their stabilization parameters.

The development and application of models of reduced computational complexity is used extensively throughout science and engineering to enable the fast/real-time modeling of complex systems for control, design, or prediction purposes. These models, while often successful and of undisputed value, are, however, often heuristic in nature and the validity and accuracy of the output is often unknown. This limits the predictive value of such models.

In this tutorial we will review recent and ongoing efforts to develop reduced basis methods for which one can develop a rigorous a posteriori theory. The approach aims at formulating reduced models for parameterized linear partial differential equations. We will outline the theoretical developments of certified reduced basis methods, discuss an offline-online approach to ensure computational efficiency, and emphasize how an error estimator can be exploited to construct an efficient basis at minimal computational off-line cost. We also discuss recent improvements on the efficiency of the computation of the lower bounds for the error, using an improved Successive Constraint Method. The discussion will draw on examples based both on differential and integral equations formulations.

The performance of the certified reduced basis model will be illustrated through several examples to highlight the major advantages of the proposed approach as well as key open challenges in the current approach.

Time permitting we will extend the discussion to include problems with parameterized geometries and the introduction of reduced element methods to enable the efficient and accurate modeling of networks and geometrically complex configurations.

A broad range of scientific and engineering problems involve multiple scales. Traditional approaches have been known to be valid for limited spatial and temporal scales. Multiple scales dominate simulation efforts wherever large disparities in spatial and temporal scales are encountered. Such disparities appear in virtually all areas of modern science and engineering, for example, composite materials, porous media, turbulent transport in high Reynolds number flows, and so on. Here, we review some recent advances in multiscale finite element methods (MsFEM) and their applications. The notion ``multiscale finite element methods'' refers to a number of methods, such as multiscale finite volume, mixed multiscale finite element method, and the like. The concept that unifies these methods is the coupling of oscillatory basis functions via various variational formulations. One of the main aspects of this coupling is the subgrid capturing errors. We attempt to capture the multiscale structure of the solution via localized basis functions. These basis functions contain essential multiscale information embedded in the solution and are coupled through a global formulation to provide a faithful approximation of the solution.

The lecture will start with some basic ideas behind MsFEM and its error analysis. We will put special emphasis on how to design appropriate boundary conditions for the local bases to minimize the subgrid capturing errors. In some cases, limited global information is required to capture the long range correlation among small scales. One way to achieve this is through an iterative precodure between the global large scale solution and the localized subgrid scale solution. We will also compare MsFEM with a few related multiscale methods. Applications to high contrast interface problems, two-phase flows in strongly heterogeneous porous media, uncertainty quantification, and domain decompositions will be discussed. Finally, we will present a new data-driven stochastic multiscale method for solving stochastic PDEs, which is in part inspired by MsFEM.

I shall talk about how to design fast spectral-Galerkin algorithms for some prototypical partial differential equations. We shall start with algorithms in one dimension, then using a tensor product approach for two and three dimensions, and hyperbolic cross/spectral sparse grid for higher dimensional problems.

The purpose of this tutorial is to give an introduction to finite element exterior calculus, targeted to an audience which is reasonably familiar with topics like elliptic partial differential equations, Sobolev spaces, and finite element methods. We will first give a brief review of some of the fundamental concepts of exterior calculus, such as interior and exterior products, pullbacks, the Hodge star operation, the exterior derivative, and Stokes' theorem. Then we will focus on some of the main building blocks of finite element exterior calculus. In particular, we will discuss piecewise polynomial spaces of differential forms, degress of freedom, and the construction of bounded cochain projections. In addition, an abstract theory of Hilbert complexes will be presented, and we will explain how this relates to to the stability theory for approximations of the Hodge Laplacian.