Relevant bibliographies by topics / Least-squares finite element

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  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Least-squares finite element'

Author: Grafiati
Published: 4 June 2021
Last updated: 3 February 2022

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Journal articles on the topic "Least-squares finite element"

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Musivand-Arzanfudi, M., and H. Hosseini-Toudeshky. "Moving least-squares finite element method." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 221, no. 9 (September 1, 2007): 1019–36. http://dx.doi.org/10.1243/09544062jmes463.

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A new computational method here called moving least-squares finite element method (MLSFEM) is presented, in which the shape functions of the parametric elements are constructed using moving least-squares approximation. While preserving some excellent characteristics of the meshless methods such as elimination of the volumetric locking in near-incompressible materials and giving accurate strains and stresses near the boundaries of the problem, the computational time is decreased by constructing the meshless shape functions in the stage of creating parametric elements and then utilizing them for any new problem. Moreover, it is not necessary to have knowledge about the full details of the shape function generation method in future uses. The MLSFEM also eliminates another drawback of meshless methods associated with the lack of accordance between the integration cells and the problem boundaries. The method is described for two-dimensional problems, but it is extendable for three-dimensional problems too. The MLSFEM does not require the complex mesh generation. Excellent results can be obtained even using a simple mesh. A technique is also presented for isoparametric mapping which enables best possible mapping via a constrained optimization criterion. Several numerical examples are analysed to show the efficiency and convergence of the method.
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Keith, Brendan, Socratis Petrides, Federico Fuentes, and Leszek Demkowicz. "Discrete least-squares finite element methods." Computer Methods in Applied Mechanics and Engineering 327 (December 2017): 226–55. http://dx.doi.org/10.1016/j.cma.2017.08.043.

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Duan, Huo-Yuan, and Guo-Ping Liang. "Nonconforming elements in least-squares mixed finite element methods." Mathematics of Computation 73, no. 245 (March 27, 2003): 1–18. http://dx.doi.org/10.1090/s0025-5718-03-01520-5.

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Bochev, Pavel B., and Max D. Gunzburger. "Finite Element Methods of Least-Squares Type." SIAM Review 40, no. 4 (January 1998): 789–837. http://dx.doi.org/10.1137/s0036144597321156.

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Chen, T. F., and G. J. Fix. "Least squares finite element simulation of transonic flows." Applied Numerical Mathematics 2, no. 3-5 (October 1986): 399–408. http://dx.doi.org/10.1016/0168-9274(86)90042-5.

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Bedivan, D. M. "Error estimates for least squares finite element methods." Computers & Mathematics with Applications 43, no. 8-9 (April 2002): 1003–20. http://dx.doi.org/10.1016/s0898-1221(02)80009-8.

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Brannick, J., C. Ketelsen, T. Manteuffel, and S. McCormick. "Least-Squares Finite Element Methods for Quantum Electrodynamics." SIAM Journal on Scientific Computing 32, no. 1 (January 2010): 398–417. http://dx.doi.org/10.1137/080729633.

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Kiousis, Panos D. "Least-Squares Finite-Element Evaluation of Flow Nets." Journal of Geotechnical and Geoenvironmental Engineering 128, no. 8 (August 2002): 699–701. http://dx.doi.org/10.1061/(asce)1090-0241(2002)128:8(699).

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Bochev, Pavel, Leszek Demkowicz, Jay Gopalakrishnan, and Max Gunzburger. "Minimum Residual and Least Squares Finite Element Methods." Computers & Mathematics with Applications 68, no. 11 (December 2014): 1479. http://dx.doi.org/10.1016/j.camwa.2014.11.005.

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Chen, Fuchen, Eric Chung, and Lijian Jiang. "Least-squares mixed generalized multiscale finite element method." Computer Methods in Applied Mechanics and Engineering 311 (November 2016): 764–87. http://dx.doi.org/10.1016/j.cma.2016.09.010.

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Dissertations / Theses on the topic "Least-squares finite element"

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Wei, Fei. "Weighted least-squares finite element methods for PIV data assimilation." Thesis, Montana State University, 2011. http://etd.lib.montana.edu/etd/2011/wei/WeiF0811.pdf.

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The ability to diagnose irregular flow patterns clinically in the left ventricle (LV) is currently very challenging. One potential approach for non-invasively measuring blood flow dynamics in the LV is particle image velocimetry (PIV) using microbubbles. To obtain local flow velocity vectors and velocity maps, PIV software calculates displacements of microbubbles over a given time interval, which is typically determined by the actual frame rate. In addition to the PIV, ultrasound images of the left ventricle can be used to determine the wall position as a function of time, and the inflow and outflow fluid velocity during the cardiac cycle. Despite the abundance of data, ultrasound and PIV alone are insufficient for calculating the flow properties of interest to clinicians. Specifically, the pressure gradient and total energy loss are of primary importance, but their calculation requires a full three-dimensional velocity field. Echo-PIV only provides 2D velocity data along a single plane within the LV. Further, numerous technical hurdles prevent three-dimensional ultrasound from having a sufficiently high frame rate (currently approximately 10 frames per second) for 3D PIV analysis. Beyond microbubble imaging in the left ventricle, there are a number of other settings where 2D velocity data is available using PIV, but a full 3D velocity field is desired. This thesis develops a novel methodology to assimilate two-dimensional PIV data into a three-dimensional Computational Fluid Dynamics simulation with moving domains. To illustrate and validate our approach, we tested the approach on three different problems: a flap displaced by a fluid jut; an expanding hemisphere; and an expanding half ellipsoid representing the left ventricle of the heart. To account for the changing shape of the domain in each problem, the CFD mesh was deformed using a pseudo-solid domain mapping technique at each time step. The incorporation of experimental PIV data can help to identify when the imposed boundary conditions are incorrect. This approach can also help to capture effects that are not modeled directly like the impacts of heart valves on the flow of blood into the left ventricle.
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Bringmann, Philipp. "Adaptive least-squares finite element method with optimal convergence rates." Doctoral thesis, Humboldt-Universität zu Berlin, 2021. http://dx.doi.org/10.18452/22350.

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Die Least-Squares Finite-Elemente-Methoden (LSFEMn) basieren auf der Minimierung des Least-Squares-Funktionals, das aus quadrierten Normen der Residuen eines Systems von partiellen Differentialgleichungen erster Ordnung besteht. Dieses Funktional liefert einen a posteriori Fehlerschätzer und ermöglicht die adaptive Verfeinerung des zugrundeliegenden Netzes. Aus zwei Gründen versagen die gängigen Methoden zum Beweis optimaler Konvergenzraten, wie sie in Carstensen, Feischl, Page und Praetorius (Comp. Math. Appl., 67(6), 2014) zusammengefasst werden. Erstens scheinen fehlende Vorfaktoren proportional zur Netzweite den Beweis einer schrittweisen Reduktion der Least-Squares-Schätzerterme zu verhindern. Zweitens kontrolliert das Least-Squares-Funktional den Fehler der Fluss- beziehungsweise Spannungsvariablen in der H(div)-Norm, wodurch ein Datenapproximationsfehler der rechten Seite f auftritt. Diese Schwierigkeiten führten zu einem zweifachen Paradigmenwechsel in der Konvergenzanalyse adaptiver LSFEMn in Carstensen und Park (SIAM J. Numer. Anal., 53(1), 2015) für das 2D-Poisson-Modellproblem mit Diskretisierung niedrigster Ordnung und homogenen Dirichlet-Randdaten. Ein neuartiger expliziter residuenbasierter Fehlerschätzer ermöglicht den Beweis der Reduktionseigenschaft. Durch separiertes Markieren im adaptiven Algorithmus wird zudem der Datenapproximationsfehler reduziert. Die vorliegende Arbeit verallgemeinert diese Techniken auf die drei linearen Modellprobleme das Poisson-Problem, die Stokes-Gleichungen und das lineare Elastizitätsproblem. Die Axiome der Adaptivität mit separiertem Markieren nach Carstensen und Rabus (SIAM J. Numer. Anal., 55(6), 2017) werden in drei Raumdimensionen nachgewiesen. Die Analysis umfasst Diskretisierungen mit beliebigem Polynomgrad sowie inhomogene Dirichlet- und Neumann-Randbedingungen. Abschließend bestätigen numerische Experimente mit dem h-adaptiven Algorithmus die theoretisch bewiesenen optimalen Konvergenzraten.
The least-squares finite element methods (LSFEMs) base on the minimisation of the least-squares functional consisting of the squared norms of the residuals of first-order systems of partial differential equations. This functional provides a reliable and efficient built-in a posteriori error estimator and allows for adaptive mesh-refinement. The established convergence analysis with rates for adaptive algorithms, as summarised in the axiomatic framework by Carstensen, Feischl, Page, and Praetorius (Comp. Math. Appl., 67(6), 2014), fails for two reasons. First, the least-squares estimator lacks prefactors in terms of the mesh-size, what seemingly prevents a reduction under mesh-refinement. Second, the first-order divergence LSFEMs measure the flux or stress errors in the H(div) norm and, thus, involve a data resolution error of the right-hand side f. These difficulties led to a twofold paradigm shift in the convergence analysis with rates for adaptive LSFEMs in Carstensen and Park (SIAM J. Numer. Anal., 53(1), 2015) for the lowest-order discretisation of the 2D Poisson model problem with homogeneous Dirichlet boundary conditions. Accordingly, some novel explicit residual-based a posteriori error estimator accomplishes the reduction property. Furthermore, a separate marking strategy in the adaptive algorithm ensures the sufficient data resolution. This thesis presents the generalisation of these techniques to three linear model problems, namely, the Poisson problem, the Stokes equations, and the linear elasticity problem. It verifies the axioms of adaptivity with separate marking by Carstensen and Rabus (SIAM J. Numer. Anal., 55(6), 2017) in three spatial dimensions. The analysis covers discretisations with arbitrary polynomial degree and inhomogeneous Dirichlet and Neumann boundary conditions. Numerical experiments confirm the theoretically proven optimal convergence rates of the h-adaptive algorithm.
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Storn, Johannes. "Topics in Least-Squares and Discontinuous Petrov-Galerkin Finite Element Analysis." Doctoral thesis, Humboldt-Universität zu Berlin, 2019. http://dx.doi.org/10.18452/20141.

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Aufgrund der fundamentalen Bedeutung partieller Differentialgleichungen zur Beschreibung von Phänomenen in angewandten Wissenschaften ist deren Analyse ein Kerngebiet der Mathematik. Durch Computer lassen sich die Lösungen für eine Vielzahl dieser Gleichungen näherungsweise bestimmen. Die dabei verwendeten numerischen Verfahren sollen auf möglichst exakte Approximationen führen und deren Genauigkeit verifizieren. Die Least-Squares Finite-Elemente-Methode (LSFEM) und die unstetige Petrov-Galerkin (DPG) Methode sind solche Verfahren. Sie werden in dieser Dissertation untersucht. Der erste Teil der Arbeit untersucht die Genauigkeit der mittels LSFEM berechneten Näherungen. Dazu werden Eigenschaften der zugrundeliegenden Differentialgleichungen mit den Eigenschaften der LSFEM kombiniert. Dies zeigt, dass die Abweichung der berechneten Näherung von der exakten Lösung einem berechenbaren Residuum asymptotisch entspricht. Ferner wird ein Verfahren zu Berechnung einer garantierten oberen Fehlerschranke eingeführt. Während etablierte Fehlerschätzer den Fehler signifikant überschätzt, zeigen numerische Experimente eine äußerst geringe Überschätzung des Fehlers mittels der neuen Fehlerschranke. Die Analyse der Fehlerschranken für das Stokes-Problem offenbart ein Beziehung der LSFEM und der LBB Konstanten. Diese Konstante ist entscheidend für die Existenz und Stabilität von Lösungen in der Strömungslehre. Der zweite Teil der Arbeit nutzt diese Beziehung und entwickelt ein auf der LSFEM basierendes Verfahren zur numerischen Berechnung der LBB Konstanten. Der dritte Teil der Arbeit untersucht die DPG Methode. Dabei werden existierende Anwendungen der DPG Methode zusammengefasst und analysiert. Diese Analyse zeigt, dass sich die DPG Methode als eine leicht gestörte LSFEM interpretieren lässt. Diese Interpretation erlaubt die Anwendung der Resultate aus dem ersten Teil der Arbeit und ermöglicht dadurch eine genauere Untersuchung existierender und die Entwicklung neuer DPG Methoden.
The analysis of partial differential equations is a core area in mathematics due to the fundamental role of partial differential equations in the description of phenomena in applied sciences. Computers can approximate the solutions to these equations for many problems. They use numerical schemes which should provide good approximations and verify the accuracy. The least-squares finite element method (LSFEM) and the discontinuous Petrov-Galerkin (DPG) method satisfy these requirements. This thesis investigates these two schemes. The first part of this thesis explores the accuracy of solutions to the LSFEM. It combines properties of the underlying partial differential equation with properties of the LSFEM and so proves the asymptotic equality of the error and a computable residual. Moreover, this thesis introduces an novel scheme for the computation of guaranteed upper error bounds. While the established error estimator leads to a significant overestimation of the error, numerical experiments indicate a tiny overestimation with the novel bound. The investigation of error bounds for the Stokes problem visualizes a relation of the LSFEM and the Ladyzhenskaya-Babuška-Brezzi (LBB) constant. This constant is a key in the existence and stability of solution to problems in fluid dynamics. The second part of this thesis utilizes this relation to design a competitive numerical scheme for the computation of the LBB constant. The third part of this thesis investigates the DPG method. It analyses an abstract framework which compiles existing applications of the DPG method. The analysis relates the DPG method with a slightly perturbed LSFEM. Hence, the results from the first part of this thesis extend to the DPG method. This enables a precise investigation of existing and the design of novel DPG schemes.
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Akargun, Yigit Hayri. "Least-squares Finite Element Solution Of Euler Equations With Adaptive Mesh Refinement." Master's thesis, METU, 2012. http://etd.lib.metu.edu.tr/upload/12614138/index.pdf.

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Least-squares finite element method (LSFEM) is employed to simulate 2-D and axisymmetric flows governed by the compressible Euler equations. Least-squares formulation brings many advantages over classical Galerkin finite element methods. For non-self-adjoint systems, LSFEM result in symmetric positive-definite matrices which can be solved efficiently by iterative methods. Additionally, with a unified formulation it can work in all flight regimes from subsonic to supersonic. Another advantage is that, the method does not require artificial viscosity since it is naturally diffusive which also appears as a difficulty for sharply resolving high gradients in the flow field such as shock waves. This problem is dealt by employing adaptive mesh refinement (AMR) on triangular meshes. LSFEM with AMR technique is numerically tested with various flow problems and good agreement with the available data in literature is seen.
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Goktolga, Mustafa Ugur. "Simulation Of Conjugate Heat Transfer Problems Using Least Squares Finite Element Method." Master's thesis, METU, 2012. http://etd.lib.metu.edu.tr/upload/12614787/index.pdf.

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In this thesis study, a least-squares finite element method (LSFEM) based conjugate heat transfer solver was developed. In the mentioned solver, fluid flow and heat transfer computations were performed separately. This means that the calculated velocity values in the flow calculation part were exported to the heat transfer part to be used in the convective part of the energy equation. Incompressible Navier-Stokes equations were used in the flow simulations. In conjugate heat transfer computations, it is required to calculate the heat transfer in both flow field and solid region. In this study, conjugate behavior was accomplished in a fully coupled manner, i.e., energy equation for fluid and solid regions was solved simultaneously and no boundary conditions were defined on the fluid-solid interface. To assure that the developed solver works properly, lid driven cavity flow, backward facing step flow and thermally driven cavity flow problems were simulated in three dimensions and the findings compared well with the available data from the literature. Couette flow and thermally driven cavity flow with conjugate heat transfer in two dimensions were modeled to further validate the solver. Finally, a microchannel conjugate heat transfer problem was simulated. In the flow solution part of the microchannel problem, conservation of mass was not achieved. This problem was expected since the LSFEM has problems related to mass conservation especially in high aspect ratio channels. In order to overcome the mentioned problem, weight of continuity equation was increased by multiplying it with a constant. Weighting worked for the microchannel problem and the mass conservation issue was resolved. Obtained results for microchannel heat transfer problem were in good agreement in general with the previous experimental and numerical works. In the first computations with the solver
quadrilateral and triangular elements for two dimensional problems, hexagonal and tetrahedron elements for three dimensional problems were tried. However, since only the quadrilateral and hexagonal elements gave satisfactory results, they were used in all the above mentioned simulations.
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Johnsen, Eivind. "Application method of the least squares finite element method to fracture mechanics." Thesis, Georgia Institute of Technology, 1995. http://hdl.handle.net/1853/16435.

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Danisch, Garvin. "Gemischte Finite-element-least-squares-Methoden für die Flachwassergleichungen mit kleiner Viskosität." [S.l.] : [s.n.], 2007. http://deposit.ddb.de/cgi-bin/dokserv?idn=983833052.

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Dolan, P. S. "Viscous incompressible flow solutions via divergence free least squares finite element optimisation." Thesis, University of Hertfordshire, 1986. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.377903.

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Prabhakar, Vivek. "Least squares based finite element formulations and their applications in fluid mechanics." [College Station, Tex. : Texas A&M University, 2006. http://hdl.handle.net/1969.1/ETD-TAMU-1152.

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Bochev, Pavel B. "Least squares finite element methods for the Stokes and Navier-Stokes equations." Diss., This resource online, 1994. http://scholar.lib.vt.edu/theses/available/etd-06062008-165910/.

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Books on the topic "Least-squares finite element"

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D, Gunzburger Max, ed. Least-squares finite element methods. New York: Springer, 2009.

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Bochev, Pavel B. Least-squares finite element methods. New York: Springer, 2009.

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Gunzburger, Max D., and Pavel B. Bochev. Least-Squares Finite Element Methods. New York, NY: Springer New York, 2009. http://dx.doi.org/10.1007/b13382.

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Bochev, Pavel B. Least-squares finite element methods. New York: Springer, 2009.

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Jiang, Bo-nan. The Least-Squares Finite Element Method. Berlin, Heidelberg: Springer Berlin Heidelberg, 1998. http://dx.doi.org/10.1007/978-3-662-03740-9.

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Jiang, Bo-Nan. Least-squares finite elements for Stokes problem. Cleveland, Ohio: ICOMP, 1988.

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Jiang, Bo-nan. Least-squares finite element method for fluid dynamics. Cleveland, Ohio: Institute for Computational Mechanics in Propulsion, 1989.

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Bochev, Pavel B. Accuracy of least-squares method for the Navier-Stokes equations. [Washington, DC]: National Aeronautics and Space Administration, 1993.

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Bochev, Pavel B. Accuracy of least-squares method for the Navier-Stokes equations. [Washington, DC]: National Aeronautics and Space Administration, 1993.

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Bochev, Pavel B. Accuracy of least-squares method for the Navier-Stokes equations. [Washington, DC]: National Aeronautics and Space Administration, 1993.

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Book chapters on the topic "Least-squares finite element"

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Bochev, Pavel, and Max Gunzburger. "Least Squares Finite Element Methods." In Encyclopedia of Applied and Computational Mathematics, 782–85. Berlin, Heidelberg: Springer Berlin Heidelberg, 2015. http://dx.doi.org/10.1007/978-3-540-70529-1_330.

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Bochev, Pavel B., and Max D. Gunzburger. "Variations on Least-Squares Finite Element Methods." In Applied Mathematical Sciences, 1–56. New York, NY: Springer New York, 2009. http://dx.doi.org/10.1007/b13382_12.

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Jiang, Bonan, and Guojun Liao. "The Least-Squares Meshfree Finite Element Method." In Computational Mechanics, 341. Berlin, Heidelberg: Springer Berlin Heidelberg, 2007. http://dx.doi.org/10.1007/978-3-540-75999-7_141.

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Bochev, Pavel B., and Max D. Gunzburger. "Mathematical Foundations of Least-Squares Finite Element Methods." In Applied Mathematical Sciences, 1–33. New York, NY: Springer New York, 2009. http://dx.doi.org/10.1007/b13382_3.

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Starke, Gerhard. "Adaptive Least Squares Finite Element Methods in Elasto-Plasticity." In Large-Scale Scientific Computing, 671–78. Berlin, Heidelberg: Springer Berlin Heidelberg, 2010. http://dx.doi.org/10.1007/978-3-642-12535-5_80.

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Schröder, Jörg, Alexander Schwarz, and Karl Steeger. "Least-Squares Mixed Finite Element Formulations for Isotropic and Anisotropic Elasticity at Small and Large Strains." In Advanced Finite Element Technologies, 131–75. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-31925-4_6.

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Chen, Yanping, and Manping Zhang. "Superconvergence of Least-Squares Mixed Finite Element Approximations Over Quadrilaterals." In Recent Progress in Computational and Applied PDES, 135–44. Boston, MA: Springer US, 2002. http://dx.doi.org/10.1007/978-1-4615-0113-8_9.

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Bochev, Pavel B., and Max D. Gunzburger. "The Agmon–Douglis–Nirenberg Setting for Least-Squares Finite Element Methods." In Applied Mathematical Sciences, 1–28. New York, NY: Springer New York, 2009. http://dx.doi.org/10.1007/b13382_4.

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Zhu, Yue, and Lingmi Zhang. "Finite Element Model Updating Based on Least Squares Support Vector Machines." In Advances in Neural Networks – ISNN 2009, 296–303. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009. http://dx.doi.org/10.1007/978-3-642-01510-6_34.

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Adler, J. H., and P. S. Vassilevski. "Improving Conservation for First-Order System Least-Squares Finite-Element Methods." In Numerical Solution of Partial Differential Equations: Theory, Algorithms, and Their Applications, 1–19. New York, NY: Springer New York, 2013. http://dx.doi.org/10.1007/978-1-4614-7172-1_1.

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Conference papers on the topic "Least-squares finite element"

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Lili Li. "Least-squares finite element for nonlinear MHD equations." In 2010 International Conference on Educational and Network Technology (ICENT 2010). IEEE, 2010. http://dx.doi.org/10.1109/icent.2010.5532189.

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Liang, Shin-Jye. "A Least-Squares Finite-Element Method for Shallow-Water Equations." In OCEANS 2008 - MTS/IEEE Kobe Techno-Ocean. IEEE, 2008. http://dx.doi.org/10.1109/oceanskobe.2008.4531097.

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Hou, Lin-Jun. "A time-accurate least-squares finite element method for incompressible flow." In 33rd Aerospace Sciences Meeting and Exhibit. Reston, Virigina: American Institute of Aeronautics and Astronautics, 1995. http://dx.doi.org/10.2514/6.1995-81.

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Kumar, Rajeev, and Brian H. Dennis. "The Least-Squares Galerkin Split Finite Element Method for Buoyancy-Driven Flow." In ASME 2010 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2010. http://dx.doi.org/10.1115/detc2010-29157.

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The least-squares finite element method (LSFEM), based on minimizing the l2-norm of the residual is now well established as a proper approach to deal with the convection dominated fluid dynamic equations. The least-squares finite element method has a number of attractive characteristics such as the lack of an inf-sup condition and the resulting symmetric positive system of algebraic equations unlike Galerkin finite element method (GFEM). However, the higher continuity requirements for second-order terms in the governing equations force the introduction of additional unknowns through the use of an equivalent first-order system of equations or the use of C1 continuous basis functions. These additional unknowns lead to increased memory and computational requirements that have limited the application of LSFEM to large-scale practical problems. A novel finite element method is proposed that employs a least-squares method for first-order derivatives and a Galerkin method for second order derivatives, thereby avoiding the need for additional unknowns required by a pure LSFEM approach. When the unsteady form of the governing equations is used, a streamline upwinding term is introduced naturally by the least-squares method. Resulting system matrix is always symmetric and positive definite and can be solved by iterative solvers like pre-conditioned conjugate gradient method. The method is stable for convection-dominated flows and allows for equal-order basis functions for both pressure and velocity. The method has been successfully applied here to solve complex buoyancy-driven flow with Boussinesq approximation in a square cavity with differentially heated vertical walls using low-order C0 continuous elements.
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Rasmussen, Cody, Robert Canfield, and J. Reddy. "The Least-Squares Finite Element Method Applied to Fluid-Structure Interation Problems." In 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference. Reston, Virigina: American Institute of Aeronautics and Astronautics, 2007. http://dx.doi.org/10.2514/6.2007-2407.

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Zhengliu Zhou, Scott Keller, Abdon Sepulveda, and Greg Carman. "Simulating vibration-induced electromagnetic fields with Galerkin/least-squares finite element methods." In 2016 IEEE International Ultrasonics Symposium (IUS). IEEE, 2016. http://dx.doi.org/10.1109/ultsym.2016.7728727.

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French, Donald A., Christopher R. Schrock, and John A. Benek. "Least squares overset finite element method for scalar hyperbolic problems in 2D." In 23rd AIAA Computational Fluid Dynamics Conference. Reston, Virginia: American Institute of Aeronautics and Astronautics, 2017. http://dx.doi.org/10.2514/6.2017-4277.

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JIANG, BO-NAN, T. LIN, LIN-JUN HOU, and LOUIS POVINELLI. "A least-squares finite element method for 3D incompressible Navier-Stokes equations." In 31st Aerospace Sciences Meeting. Reston, Virigina: American Institute of Aeronautics and Astronautics, 1993. http://dx.doi.org/10.2514/6.1993-338.

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9

Kumar, Rajeev, and Brian H. Dennis. "A Least-Squares/Galerkin Finite Element Method for Incompressible Navier-Stokes Equations." In ASME 2008 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2008. http://dx.doi.org/10.1115/detc2008-49654.

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Abstract:
The least-squares finite element method (LSFEM), which is based on minimizing the l2-norm of the residual, has many attractive advantages over Galerkin finite element method (GFEM). It is now well established as a proper approach to deal with the convection dominated fluid dynamic equations. The least-squares finite element method has a number of attractive characteristics such as the lack of an inf-sup condition and the resulting symmetric positive system of algebraic equations unlike GFEM. However, the higher continuity requirements for second-order terms in the governing equations force the introduction of additional unknowns through the use of an equivalent first-order system of equations or the use of C1 continuous basis functions. These additional unknowns lead to increased memory and computing time requirements that have prevented the application of LSFEM to large-scale practical problems, such as three-dimensional compressible viscous flows. A simple finite element method is proposed that employs a least-squares method for first-order derivatives and a Galerkin method for second order derivatives, thereby avoiding the need for additional unknowns required by pure a LSFEM approach. When the unsteady form of the governing equations is used, a streamline upwinding term is introduced naturally by the leastsquares method. Resulting system matrix is always symmetric and positive definite and can be solved by iterative solvers like pre-conditioned conjugate gradient method. The method is stable for convection-dominated flows and allows for equalorder basis functions for both pressure and velocity. The stability and accuracy of the method are demonstrated with preliminary results of several benchmark problems solved using low-order C0 continuous elements.
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10

Kumar, Rajeev, and Brian H. Dennis. "A Least-Squares Galerkin Split Finite Element Method for Compressible Navier-Stokes Equations." In ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2009. http://dx.doi.org/10.1115/detc2009-87569.

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Abstract:
A novel finite element method is proposed that employs a least-squares method for first-order derivatives and a Galerkin method for second order derivatives, thereby avoiding the need for additional unknowns required by a pure LSFEM approach. When the unsteady form of the governing equations is used, a streamline upwinding term is introduced naturally by the least-squares method. Resulting system matrix is always symmetric and positive definite and can be solved by iterative solvers like pre-conditioned conjugate gradient method. The method is stable for convection-dominated flows and allows for equal-order basis functions for both pressure and velocity. The stability and accuracy of the method are demonstrated in the context of compressible flows by results of few compressible benchmark problems solved using low-order C0 continuous elements.
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