The DPG method for the Stokes problem

Roberts, Nathan V.; Bui-Thanh, Tan; Demkowicz, Leszek

doi:10.1016/j.camwa.2013.12.015

Cited by 49 publications

(48 citation statements)

References 18 publications

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“…In choosing the best algorithm to solve the least-squares problem coming from a DLS method, many factors are important to consider. For instance, the normal equation have been demonstrated to be adequate when the methodology has been applied to many DPG problems [64,65,21,67,33,37,49,62,30,40,38,35]. Indeed, in many reasonable circumstances, the round-off error in the solution from the associated linear solve cannot be expected to be nearly as large as the truncation error due to the finite element discretization.…”

Section: Solution Algorithmsmentioning

confidence: 99%

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Discrete least-squares finite element methods

Keith

Petrides

Fuentes

et al. 2017

Computer Methods in Applied Mechanics and Engineering

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ABSTRACT. A finite element methodology for large classes of variational boundary value problems is defined which involves discretizing two linear operators: (1) the differential operator defining the spatial boundary value problem; and (2) a Riesz map on the test space. The resulting linear system is overdetermined. Two different approaches for solving the system are suggested (although others are discussed): (1) solving the associated normal equation with linear solvers for symmetric positive-definite systems (e.g. Cholesky factorization); and (2) solving the overdetermined system with orthogonalization algorithms (e.g. QR factorization). The finite element assembly algorithm for each of these approaches is described in detail. The normal equation approach is usually faster for direct solvers and requires less storage. The second approach reduces the condition number of the system by a power of two and is less sensitive to round-off error. The rectangular stiffness matrix of second approach is demonstrated to have condition number O(h −1 ) for a variety of formulations of Poisson's equation. The stiffness matrix from the normal equation approach is demonstrated to be related to the monolithic stiffness matrices of least-squares finite element methods and it is proved that the two are identical in some cases. An example with Poisson's equation indicates that the solutions of these two different linear systems can be nearly indistinguishable (if round-off error is not an issue) and rapidly converge to each other. The orthogonalization approach is suggested to be beneficial for problems which induce poorly conditioned linear systems. Experiments with Poisson's equation in single-precision arithmetic as well as the linear acoustics problem near resonance in double-precision arithmetic verify this conclusion. The methodology described here was developed as an outgrowth of the discontinuous Petrov-Galerkin (DPG) methodology of Demkowicz and Gopalakrishnan [29]. The strength of DPG is highlighted throughout, however, the majority of theory presented is general. Extensions to constrained minimization principles are also considered throughout but are not analyzed in experiments.

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Section: Solution Algorithmsmentioning

confidence: 99%

“…Because G can be efficiently inverted with DPG, in that setting, much smaller problems, posed solely in the primal variable u, can be solved directly. This has been performed for many problems of engineering interest [65,21,67,33,37,49,62,30,40,38,35].…”

Section: Introductionmentioning

confidence: 99%

Discrete least-squares finite element methods

Keith

Petrides

Fuentes

et al. 2017

Computer Methods in Applied Mechanics and Engineering

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“…We first consider the ultraweak variational formulation [44][45][46] that can be derived from (12) by integration by parts. Instead of shifting the derivative on the solution u h as in (13), the ultra-weak form shifts all derivatives to the test function δu h .…”

Section: Symmetrization By Averaging With the Ultra-weak Formulationmentioning

confidence: 99%

“…If we choose approximation and test functions based on the same GLL basis in the sense of a Bubnov-Galerkin method, the symmetry of stiffness matrices in hp-collocation can be restored also for non-affine elements. To this end, we average the weighted residual formulation with a dual variational formulation based on the ultra-weak formulation [44][45][46]. Symmetry is essential for reducing memory, speeding up formation and assembly procedures, and for the application of highly efficient iterative solvers based on conjugate gradients.…”

Section: Introductionmentioning

confidence: 99%

A collocated C⁰ finite element method: Reduced quadrature perspective, cost comparison with standard finite elements, and explicit structural dynamics

Schillinger

Evans

Frischmann

et al. 2014

Numerical Meth Engineering

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We demonstrate the potential of collocation methods for efficient higher-order analysis on standard nodal finite element meshes. We focus on a collocation method that is variationally consistent and geometrically flexible, converges optimally, embraces concepts of reduced quadrature, and leads to symmetric stiffness and diagonal consistent mass matrices. At the same time, it minimizes the evaluation cost per quadrature point, thus reducing formation and assembly effort significantly with respect to standard Galerkin finite element methods. We provide a detailed review of all components of the technology in the context of elastodynamics, that is, weighted residual formulation, nodal basis functions on Gauss-Lobatto quadrature points, and symmetrization by averaging with the ultra-weak formulation. We quantify potential gains by comparing the computational efficiency of collocated and standard finite elements in terms of basic operation counts and timings. Our results show that collocation is significantly less expensive for problems dominated by the formation and assembly effort, such as higher-order elastostatic analysis. Furthermore, we illustrate the potential of collocation for efficient higher-order explicit dynamics. Throughout this work, we advocate a straightforward implementation based on simple modifications of standard finite element codes. We also point out the close connection to spectral element methods, where many of the key ideas are already established. AbstractWe demonstrate the potential of collocation methods for efficient higher-order analysis on standard nodal finite element meshes. We focus on a collocation method that is variationally consistent and geometrically flexible, converges optimally, embraces concepts of reduced quadrature, and leads to symmetric stiffness and diagonal consistent mass matrices. At the same time, it minimizes the evaluation cost per quadrature point, thus reducing formation and assembly effort significantly with respect to standard Galerkin finite element methods. We provide a detailed review of all components of the technology in the context of elastodynamics, i.e. weighted residual formulation, nodal basis functions on Gauss-Lobatto quadrature points, and symmetrization by averaging with the ultra-weak formulation. We quantify potential gains by comparing the computational efficiency of collocation and standard finite elements in terms of basic operation counts and timings. Our results show that collocation is significantly less expensive for problems dominated by the formation and assembly effort, such as higher-order elastostatic analysis. Furthermore we illustrate the potential of collocation for efficient higher-order explicit dynamics. Throughout this work, we advocate a straightforward implementation based on simple modifications of standard finite element codes. We also point out the close connection to spectral element methods, where many of the key ideas are already established.

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“…In recent years, the discontinuous Petrov-Galerkin method with optimal test functions ("DPG method" in the following) has proved to be an attractive strategy to produce infsup stable approximations for a wide class of problems. The basic setting stems from Demkowicz and Gopalakrishnan [14,13] and has been extended, e.g., to linear elasticity [1,18], the Stokes and Maxwell equations [28,7], the Schrödinger equation [15], boundary integral and fractional equations [24,17]. Another promising application area is singularly perturbed problems [16,9,3,4,25].…”

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confidence: 99%

A DPG Framework for Strongly Monotone Operators

Cantin

Heuer

2018

SIAM J. Numer. Anal.

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Abstract. We present and analyze a hybrid technique to numerically solve strongly monotone nonlinear problems by the discontinuous Petrov-Galerkin method with optimal test functions (DPG). Our strategy is to relax the nonlinear problem to a linear one with additional unknown and to consider the nonlinear relation as a constraint. We propose to use optimal test functions only for the linear problem and to enforce the nonlinear constraint by penalization. In fact, our scheme can be seen as a minimum residual method with nonlinear penalty term. We develop an abstract framework of the relaxed DPG scheme and prove under appropriate assumptions the well-posedness of the continuous formulation and the quasi-optimal convergence of its discretization. As an application we consider an advection-diffusion problem with nonlinear diffusion of strongly monotone type. Some numerical results in the lowest-order setting are presented to illustrate the predicted convergence.Key words. Discontinuous Petrov-Galerkin method, optimal test functions, strongly monotone operator, advectiondiffusion, nonlinear penalty.AMS subject classifications. 65N30, 65J15, 65N12, 47H051. Introduction. In recent years, the discontinuous Petrov-Galerkin method with optimal test functions ("DPG method" in the following) has proved to be an attractive strategy to produce infsup stable approximations for a wide class of problems. The basic setting stems from Demkowicz and Gopalakrishnan [14,13] and has been extended, e.g., to linear elasticity [1,18], the Stokes and Maxwell equations [28,7], the Schrödinger equation [15], boundary integral and fractional equations [24,17]. Another promising application area is singularly perturbed problems [16,9,3,4,25].All the cited references, however, deal with linear problems. An extension of the DPG technology to nonlinear problems, on the other hand, is a delicate issue. Principal problem is that the calculation (or approximation) of optimal test functions involves an application of the underlying operator (the DPG method is a minimum residual method). For nonlinear problems this step thus becomes nonlinear, i.e., expensive. One way to circumvent the nonlinearity is, of course, to linearize the underlying problem. This has been the approach in [8,29]. A different idea is to apply the minimum residual technique in product or "broken" spaces to the nonlinear problem. Bui-Thanh and Ghattas [5] did this by considering the entire nonlinear problem as a constraint, and Carstensen et al.[6] developed a representation of the DPG scheme by a nonlinear mixed form and analyzed the case of lowest order approximations. DPG for contact problems has been studied in [21], though in this case the nonlinearity is due to the contact condition which is treated by a variational inequality. We also note that Muga and van der Zee [27] study problems posed in Banach spaces. In those cases the calculation of optimal test functions becomes nonlinear even though the underlying PDE is linear.In this paper we propose a combined scheme that empl...

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The DPG method for the Stokes problem

Cited by 49 publications

References 18 publications

Discrete least-squares finite element methods

Discrete least-squares finite element methods

A collocated C⁰ finite element method: Reduced quadrature perspective, cost comparison with standard finite elements, and explicit structural dynamics

A DPG Framework for Strongly Monotone Operators

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The DPG method for the Stokes problem

Cited by 49 publications

References 18 publications

Discrete least-squares finite element methods

Discrete least-squares finite element methods

A collocated C0 finite element method: Reduced quadrature perspective, cost comparison with standard finite elements, and explicit structural dynamics

A DPG Framework for Strongly Monotone Operators

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A collocated C⁰ finite element method: Reduced quadrature perspective, cost comparison with standard finite elements, and explicit structural dynamics