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We present and study a parameter-free stabilized finite element method for the scalar advection-diffusion equation. Edge element lifting of diffusive edge fluxes defines the stabilization. The amount of edge diffusion varies with the edge Peclet number and adapts to solution features without mesh-dependent parameters. We prove that the method is first-order accurate in the advective limit. Numerical studies confirm the theoretical estimates.

We present a new optimization-based method for atomistic-to-continuum (AtC) coupling. The main idea is to cast the latter as a constrained optimization problem with virtual Dirichlet controls on the interfaces between the atomistic and continuum subdomains. The optimization objective is to minimize the error between the atomistic and continuum solutions on the overlap between the two subdomains, while the atomistic and continuum force balance equations provide the constraints. Separation, rather then blending of the atomistic and continuum problems, and their subsequent use as constraints in the optimization problem distinguishes our approach from the existing AtC formulations. We present and analyze the method in the context of a one-dimensional chain of atoms modeled using a linearized two-body potential with next-nearest neighbor interactions.

We develop and analyze least-squares finite element methods for two complementary div-curl elliptic boundary value problems. The first one prescribes the tangential component of the vector field on the boundary and is solved using curl-conforming elements. The second problem specifies the normal component of the vector field and is handled by div-conforming elements. We prove that both least-squares formulations are norm-equivalent with respect to suitable discrete norms, yield optimal asymptotic error estimates, and give rise to algebraic systems that can be solved by efficient algebraic multigrid methods. Numerical results that illustrate scalability of iterative solvers and optimal rates of convergence are also included.

We develop and analyze an optimization-based approach for the robust and efficient solution of PDE problems consisting of multiple physics operators with fundamentally different mathematical properties. Our approach relies on three essential steps: decomposition of the original problem into subproblems for which robust solution algorithms are available; integration of the subproblems into an equivalent PDE-constrained optimization problem; and solution of the resulting optimization problem either directly as a fully coupled algebraic system or in the null space of the PDE constraints. This strategy gives rise to a general approach for synthesizing robust solvers for complex coupled problems from solvers for their simpler physics components.

A recent study [D. N. Arnold, D. Boffi, and R. S. Falk, SIAM J. Numer. Anal., 42 (2005), pp. 2429-2451] reveals that convergence of finite element methods using $H(\mathrm{div}\,,\Omega)$-compatible finite element spaces deteriorates on nonaffine quadrilateral grids. This phenomena is particularly troublesome for the lowest-order Raviart-Thomas elements, because it implies loss of convergence in some norms for finite element solutions of mixed and least-squares methods. In this paper we propose reformulation of finite element methods, based on the natural mimetic divergence operator [M. Shashkov, Conservative Finite Difference Methods on General Grids, CRC Press, Boca Raton, FL, 1996], which restores the order of convergence. Reformulations of mixed Galerkin and least-squares methods for the Darcy equation illustrate our approach. We prove that reformulated methods converge optimally with respect to a norm involving the mimetic divergence operator. Furthermore, we prove that standard and reformulated versions of the mixed Galerkin method lead to identical linear systems, but the two versions of the least-squares method are veritably different. The surprising conclusion is that the degradation of convergence in the mixed method on nonaffine quadrilateral grids is superficial, and that the lowest-order Raviart-Thomas elements are safe to use in this method. However, the breakdown in the least-squares method is real, and there one should use our proposed reformulation.

The approximate solution of optimization and optimal control problems for systems governed by linear, elliptic partial differential equations is considered. Such problems are most often solved using methods based on applying the Lagrange multiplier rule to obtain an optimality system consisting of the state system, an adjoint-state system, and optimality conditions. Galerkin methods applied to this system result in indefinite matrix problems. Here, we consider using modern least-squares finite element methods for the solution of the optimality systems. The matrix equations resulting from this approach are symmetric and positive definite and are readily amenable to uncoupling strategies. This is an important advantage of least-squares principles as they allow for a more efficient computational solution of the optimization problem. We develop an abstract theory that includes optimal error estimates for least-squares finite element methods applied to optimality systems. We then provide an application of the theory to optimization problems for the Stokes equations.

We present a new family of stabilized methods for the Stokes problem. The focus of the paper is on the lowest order velocity-pressure pairs. While not LBB compliant, their simplicity and attractive computational properties make these pairs a popular choice in engineering practice. Our stabilization approach is motivated by terms that characterize the LBB "deficiency" of the unstable spaces. The stabilized methods are defined by using these terms to modify the saddle-point Lagrangian associated with the Stokes equations. The new stabilized methods offer a number of attractive computational properties. In contrast to other stabilization procedures, they are parameter free, do not require calculation of higher order derivatives or edge-based data structures, and always lead to symmetric linear systems. Furthermore, the new methods are unconditionally stable, achieve optimal accuracy with respect to solution regularity, and have simple and straightforward implementations. We present numerical results in two and three dimensions that showcase the excellent stability and accuracy of the new methods.

Least-squares finite element methods for first-order formulations of the Poisson equation are not subject to the inf-sup condition and lead to stable solutions even when all variables are approximated by equal-order continuous finite element spaces. For such elements, one can also prove optimal convergence in the "energy" norm (equivalent to a norm on $H^{1}(\Omega)\times H(\Omega,\mbox{\rm div})$) for all variables and optimal L2 convergence for the scalar variable. However, showing optimal L2 convergence for the flux has proven to be impossible without adding the redundant curl equation to the first-order system of partial differential equations. In fact, numerical evidence strongly suggests that nodal continuous flux approximations do not posses optimal L2 accuracy. In this paper, we show that optimal L2 error rates for the flux can be achieved without the curl constraint, provided that one uses the div-conforming family of Brezzi--Douglas--Marini or Brezzi--Douglas--Duran--Fortin elements. Then, we proceed to reveal an interesting connection between a least-squares finite element method involving $H(\Omega,\mbox{\rm div})$-conforming flux approximations and mixed finite element methods based on the classical Dirichlet and Kelvin principles. We show that such least-squares finite element methods can be obtained by approximating, through an L2 projection, the Hodge operator that connects the Kelvin and Dirichlet principles. Our principal conclusion is that when implemented in this way, a least-squares finite element method combines the best computational properties of finite element methods based on each of the classical principles.

The pressure-Poisson stabilized Galerkin method for the Stokes equation requires the choice of a positive parameter. Existing theoretical predictions for the range of parameter values that yield stable discretizations seem to be very pessimistic when compared to the computational evidence. Motivated by this wide gap, we first examine a continuous prototype for this class of schemes. We show that the prototype is absolutely stable; i.e., it is stable for all parameter values, and is optimally accurate. We then define a new, practical variant of the well-known pressure-Poisson stabilized scheme. We prove that the new method is absolutely stable just like its continuous prototype and that it achieves optimal convergence rates with respect to the same mesh-independent norms. The new method differs from the standard pressure-Poisson stabilized method in several important aspects. First, its definition does not degrade to a penalty formulation for the lowest order nodal spaces. Second, the method is absolutely stable with respect to the natural norm for the problem, while the standard pressure-Poisson stabilized method is stable with respect to a mesh-dependent norm.

This paper continues the development of the least-squares methodology for the solution of the incompressible Navier--Stokes equations started in Part I. Here we again use a velocity-flux first-order Navier--Stokes system, but our focus now is on a practical algorithm based on a discrete negative norm.

This paper develops a least-squares approach to the solution of the incompressible Navier--Stokes equations in primitive variables. As with our earlier work on Stokes equations, we recast the Navier--Stokes equations as a first-order system by introducing a velocity-flux variable and associated curl and trace equations. We show that a least-squares principle based on L2 norms applied to this system yields optimal discretization error estimates in the H1 norm in each variable, including the velocity flux. An analogous principle based on the use of an H-1 norm for the reduced system (with no curl or trace constraints) is shown to yield similar estimates, but now in the L2 norm for velocity-flux and pressure. Although the H-1 least-squares principle does not allow practical implementation, these results are critical to the analysis of a practical least-squares method for the reduced system based on a discrete equivalent of the negative norm. A practical method of this type is the subject of a companion paper. Finally, we establish optimal multigrid convergence estimates for the algebraic system resulting from the L2 norm approach.

In this paper we study finite element methods of least-squares type for the stationary, incompressible Navier--Stokes equations in two and three dimensions. We consider methods based on velocity-vorticity-pressure form of the Navier--Stokes equations augmented with several nonstandard boundary conditions. Least-squares minimization principles for these boundary value problems are developed with the aid of the Agmon--Douglis--Nirenberg (ADN) elliptic theory. Among the main results of this paper are optimal error estimates for conforming finite element approximations and analysis of some nonstandard boundary conditions. Results of several computational experiments with least-squares methods which illustrate, among other things, the optimal convergence rates are also reported.