Finite Element Methods of Least-Squares Type

Bochev, Pavel B.; Gunzburger, Max

doi:10.1137/s0036144597321156

Cited by 284 publications

(242 citation statements)

References 108 publications

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“…To simulate viscoelastic flow problems, it is convenient to split the extra-stress tensor into viscous and elastic parts by changing of variables in the LS formulations [1], [2]. The LS functional has been reported to offer several theoretical and computational advantages [3], and also provides a local and inexpensive a posteriori error estimator that may be used to guide adaptive refinement [4] and is perfectly effective and reliable for error control [5]. Therefore, difficulties in solving non-Newtonian flow problems include computational limitations and singularities caused by the high number of unknowns and geometric discontinuities, respectively.…”

Section: Introductionmentioning

confidence: 99%

Numerical Simulations of Viscoelastic Fluid Flows Using a Least-Squares Finite Element Method Based on Von Mises Stress Criteria

Lee¹

2017

IJAPM

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Abstract:The work concerns least-squares (LS) finite element solutions of the Oldroyd-B viscoelastic fluid flows using adaptive grids. To capture the viscoelastic flow region, adaptive grids are automatically generated on the basis of the von Mises stress from stress component transfer functions, and these generated refined grids agree with the physical flow attributes. Model problems considered are the flow past a slot channel problems. Numerical solutions of the flow pass through a slot channel indicate that the viscoelastic polymer solution flow characteristics are described by the refinement results of the von Mises stress. In additions, adaptive grids using the von Mises stress outperform those using functions of velocity, and satisfactory results are obtained using a lower total numbers of elements. The developed method is effective for analyzing viscoelastic fluid flows. Furthermore, the effects of low Weissenberg numbers are also investigated.

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Section: Introductionmentioning

confidence: 99%

Numerical Simulations of Viscoelastic Fluid Flows Using a Least-Squares Finite Element Method Based on Von Mises Stress Criteria

Lee¹

2017

IJAPM

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“…Examples of such alternative methods include the diffuse element or element-free Galerkin methods [9,10], least-squares FEM [11], generalized or meshless finite different methods [12,13,14,15,16], generalized or extended FEM [17,18], and partition-of-unity FEM [19]. To reduce the dependency on mesh quality, these methods avoid the use of the piecewise-polynomial Lagrange basis functions found in the classical FEM.…”

Section: Introductionmentioning

confidence: 99%

Overcoming element quality dependence of finite elements with adaptive extended stencil FEM (AES‐FEM)

Conley

Delaney

Jiao

2016

Numerical Meth Engineering

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The finite element methods (FEM) are important techniques in engineering for solving partial differential equations, but they depend heavily on element shape quality for stability and good performance. In this paper, we introduce the Adaptive Extended Stencil Finite Element Method (AES-FEM) as a means for overcoming this dependence on element shape quality. Our method replaces the traditional basis functions with a set of generalized Lagrange polynomial (GLP) basis functions, which we construct using local weighted least-squares approximations. The method preserves the theoretical framework of FEM, and allows imposing essential boundary conditions and integrating the stiffness matrix in the same way as the classical FEM. In addition, AES-FEM can use higher-degree polynomial basis functions than the classical FEM, while virtually preserving the sparsity pattern of the stiffness matrix. We describe the formulation and implementation of AES-FEM, and analyze its consistency and stability. We present numerical experiments in both 2D and 3D for the Poisson equation and a time-independent convection-diffusion equation. The numerical results demonstrate that AES-FEM is more accurate than linear FEM, is also more efficient than linear FEM in terms of error versus runtime, and enables much better stability and faster convergence of iterative solvers than linear FEM over poor-quality meshes.

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“…For a use of finite element method, the least-squares approach was widely studied in [1,2,4,5,6,13,14,19,24] and a least-squares method using pseudo spectral approximation were studied in [16,17,18,25]. The least-squares methods have several benefits such that the resulting algebraic system is always symmetric positive definite and the methods can avoid LBB compatibility condition.…”

Section: Introductionmentioning

confidence: 99%

Least-Squares Spectral Collocation Parallel Methods for Parabolic Problems

Seo¹,

Shin²

2015

Honam Mathematical Journal

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Abstract. In this paper, we study the first-order system leastsquares (FOSLS) spectral method for parabolic partial differential equations. There were lots of least-squares approaches to solve elliptic partial differential equations using finite element approximation. Also, some approaches using spectral methods have been studied in recent. In order to solve the parabolic partial differential equations in parallel, we consider a parallel numerical method based on a hybrid method of the frequency-domain method and first-order system least-squares method. First, we transform the parabolic problem in the space-time domain to the elliptic problems in the space-frequency domain. Second, we solve each elliptic problem in parallel for some frequencies using the first-order system least-squares method. And then we take the discrete inverse Fourier transforms in order to obtain the approximate solution in the space-time domain. We will introduce such a hybrid method and then present a numerical experiment.

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Finite Element Methods of Least-Squares Type

Cited by 284 publications

References 108 publications

Numerical Simulations of Viscoelastic Fluid Flows Using a Least-Squares Finite Element Method Based on Von Mises Stress Criteria

Numerical Simulations of Viscoelastic Fluid Flows Using a Least-Squares Finite Element Method Based on Von Mises Stress Criteria

Overcoming element quality dependence of finite elements with adaptive extended stencil FEM (AES‐FEM)

Least-Squares Spectral Collocation Parallel Methods for Parabolic Problems

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