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• W4237886007 abstract "International Journal of Computational Engineering ScienceVol. 04, No. 04, pp. 737-812 (2003) No AccessTHE k-VERSION OF FINITE ELEMENT METHOD FOR NON-SELF-ADJOINT OPERATORS IN BVPK. S. SURANA, A. R. AHMADI, and J. N. REDDYK. S. SURANADepartment of Mechanical Engineering, University of Kansas, Lawrence, KS 66044, USA Search for more papers by this author , A. R. AHMADIDepartment of Mechanical Engineering, University of Kansas, Lawrence, KS 66044, USA Search for more papers by this author , and J. N. REDDYDepartment of Mechanical Engineering, Texas A&M University, College Station, TX 77843-3123, USA Search for more papers by this author https://doi.org/10.1142/S1465876303002179Cited by:35 Next AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail AbstractIn a recent paper [1], the concept of the k-version finite element method was introduced for boundary value problems described by self-adjoint differential operators using the Ĥk,p spaces. Numerical results using the Galerkin method with weak form (henceforth also referred to as the Galerkin method) and least-squares finite element models were presented. In this paper, we extend the results in [1] to boundary value problems described by non-self-adjoint differential operators. We consider both the Galerkin and least-squares finite element formulations. It is shown that solutions of higher-order global smoothness or differentiability than C0 are essential in both the Galerkin and least-squares processes, and hence, use of Ĥk,p spaces is inevitable. It is demonstrated that use of the Galerkin method leads to integral or weak forms that are variationally inconsistent, whereas the least-squares processes (LSP) are variationally consistent and hence ensure non-spuriousness of the solution for all discretizations and all admissible basis functions. The least-squares processes in Ĥk,p spaces using strong form of the governing differential equations provide an mathematically consistent computational framework in which one could compute numerical solutions of any desired degree of global differentiability with high accuracy without resorting to upwinding techniques. It is shown that incorporating control over the degree of global differentiability in the computational process is absolutely essential in applications where localized high gradients of the solution are present. Numerical studies are presented using one-dimensional steady state convection diffusion equation as a model problem, for which the theoretical solution is known and is analytic, to demonstrate various features of the proposed mathematical and computational framework. Many issues related to upwinding methods such as SUPG/DC/LS and the explanation of commonly encountered problems in the Galerkin method for non-self-adjoint operators are discussed in detail. References G. Strang , Variational Crimes in Finite Element Method , Annals of Mathematics Study 33 ( Princeton University Press , Princeton, NJ , 1954 ) . Google ScholarK. S. Surana, A. R. Ahmadi and J. N. Reddy, Int. J. Comp. Engng. Sci. 3(3), 000 (2002). Google Scholar G. F. Carey and J. T. 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N. Jiang , The Least Squares Finite Element Method ( Springer-Verlag , New York , 1998 ) . Crossref, Google Scholar FiguresReferencesRelatedDetailsCited By 35k-Version of Finite Element Method for BVPs and IVPsKaran S. Surana, Celso H. Carranza and Sri Sai Charan Mathi9 June 2021 | Mathematics, Vol. 9, No. 12A Thermodynamically Consistent Formulation for Dynamic Response of Thermoviscoelastic Plate/Shell Based on Classical Continuum Mechanics (CCM)K. S. Surana and S. S. C. Mathi31 December 2020 | International Journal of Structural Stability and Dynamics, Vol. 20, No. 14Highly accurate space-time coupled least-squares finite element framework in studying wave propagationM. A. Saffarian, A. R. Ahmadi and M. H. Bagheripour16 March 2020 | SN Applied Sciences, Vol. 2, No. 4Automatic Variationally Stable Analysis for FE Computations: An IntroductionVictor M. Calo, Albert Romkes and Eirik Valseth12 August 2020Finite Element Processes Based on GM/WF in Non-Classical Solid MechanicsK. S. Surana, R. Shanbhag and J. N. Reddy1 Jan 2017 | American Journal of Computational Mathematics, Vol. 07, No. 03Non-Self-Adjoint Differential OperatorsKaran S. Surana and J. N. Reddy17 Nov 2016Error Estimations, Error Computations, and Convergence Rates in FEM for BVPsKaran S. Surana, A. D. Joy and J. N. Reddy1 Jan 2016 | Applied Mathematics, Vol. 07, No. 12Ordered rate constitutive theories in Lagrangian description for thermoviscoelastic solids with memoryK. S. Surana, T. Moody and J. N. Reddy11 June 2014 | Acta Mechanica, Vol. 226, No. 1Nonlinear Waves in Solid Continua with Finite DeformationK. S. Surana, J. Knight and J. N. Reddy1 Jan 2015 | American Journal of Computational Mathematics, Vol. 05, No. 03Mathematical models for fluid–solid interaction and their numerical solutionsK.S. Surana, B. Blackwell, M. Powell and J.N. Reddy1 Oct 2014 | Journal of Fluids and Structures, Vol. 50Riemann shock tube: 1D normal shocks in air, simulations and experimentsK.S. Surana, K.P.J. Reddy, A.D. Joy and J.N. Reddy12 June 2014 | International Journal of Computational Fluid Dynamics, Vol. 28, No. 6-10Ordered rate constitutive theories in Lagrangian description for thermoviscoelastic solids without memoryK. S. Surana, T. Moody and J. N. Reddy4 June 2013 | Acta Mechanica, Vol. 224, No. 11Computations of Evolutions for Isothermal Viscous and Viscoelastic Flows in Open DomainsK. S. Surana, Y. T. Ma, J. N. Reddy and A. Romkes1 Oct 2012 | International Journal for Computational Methods in Engineering Science and Mechanics, Vol. 13, No. 6Fluid-Solid Interaction of Incompressible Media Using h , p , k Mathematical and Computational FrameworkK. S. Surana, Y. T. Ma, J. N. Reddy and A. Romkes1 Aug 2012 | International Journal for Computational Methods in Engineering Science and Mechanics, Vol. 13, No. 5Static deflection analysis of flexural rectangular micro-plate using higher continuity finite-element methodAli Reza Ahmadi and Hamed Farahmand6 November 2012 | Mechanics & Industry, Vol. 13, No. 4Methods of Approximation in hpk Framework for ODEs in Time Resulting from Decoupling of Space and Time in IVPsK.S. Surana, L. Euler, J.N. Reddy and A. Romkes1 Jan 2011 | American Journal of Computational Mathematics, Vol. 01, No. 02Development of Mathematical Models and Computational Framework for Multi-physics Interaction ProcessesKaran S. Surana, Yongting Ma, Albert Romkes and J. N. Reddy19 Oct 2010 | Mechanics of Advanced Materials and Structures, Vol. 17, No. 7The Rate Constitutive Equations and Their Validity for Progressively Increasing DeformationKaran S. Surana, Yongting Ma, Albert Romkes and J. N. Reddy19 Oct 2010 | Mechanics of Advanced Materials and Structures, Vol. 17, No. 7Computations of Numerical Solutions in Polymer Flows Using Giesekus Constitutive Model in the hpk Framework with Variationally Consistent Integral FormsKaran S. Surana, Kedar M. Deshpande, Albert Romkes and J. N. Reddy13 Aug 2009 | International Journal for Computational Methods in Engineering Science and Mechanics, Vol. 10, No. 5J-Integral for Mode I Linear Elastic Fracture Mechanics in h, p, k Mathematical and Computational FrameworkD. Nunez, K. S. Surana, A. Romkes and J. N. Reddy13 Aug 2009 | International Journal for Computational Methods in Engineering Science and Mechanics, Vol. 10, No. 5Numerical Simulations of BVPs and IVPs in Fiber Spinning Using Giesekus Constitutive Model in hpk FrameworkKaran S. Surana, Kedar M. Deshpande, Albert Romkes and J. N. Reddy10 Mar 2009 | International Journal for Computational Methods in Engineering Science and Mechanics, Vol. 10, No. 2Numerical Solutions of BVPs in 2-D Viscous Compressible Flows Using hpk FrameworkS. Allu, K. S. Surana, A. Romkes and J. N. Reddy10 Mar 2009 | International Journal for Computational Methods in Engineering Science and Mechanics, Vol. 10, No. 2Higher Order Global Differentiability Local Approximations for 2-D Distorted Quadrilateral ElementsA. Ahmadi, K. S. Surana, R. K. Maduri, A. Romkes and J. N. Reddy12 Feb 2009 | International Journal for Computational Methods in Engineering Science and Mechanics, Vol. 10, No. 1k-Version of finite element method in 2D-polymer flows: Upper convected Maxwell modelK.S. Surana, S. Bhola, J.N. Reddy and P.W. TenPas1 Sep 2008 | Computers & Structures, Vol. 86, No. 17-18Least‐squares finite element processes in h, p, k mathematical and computational framework for a non‐linear conservation lawK. S. Surana, S. Allu, J. N. Reddy and P. W. Tenpas10 Aug 2008 | International Journal for Numerical Methods in Fluids, Vol. 57, No. 10Strong and Weak Form of the Governing Differential Equations in Least Squares Finite Element Processes in h,p,k FrameworkK. S. Surana, L. R. Anthoni, S. Allu and J. N. Reddy1 Jan 2008 | International Journal for Computational Methods in Engineering Science and Mechanics, Vol. 9, No. 1Galerkin/Least-Squares Finite Element Processes for BVP in h, p, k Mathematical FrameworkK. S. Surana, R. Kanti Mahanthi and J. N. Reddy5 Oct 2007 | International Journal for Computational Methods in Engineering Science and Mechanics, Vol. 8, No. 6Galerkin and Least-Squares Finite Element Processes for 2-D Helmholtz Equation in h , p , k FrameworkK. S. Surana, P. Gupta and J. N. Reddy31 Jul 2007 | International Journal for Computational Methods in Engineering Science and Mechanics, Vol. 8, No. 5k -Version Least Squares Finite Element Processes for 2-D Generalized Newtonian Fluid FlowsK. S. Surana, M. K. Engelkemier, J. N. Reddy and P. W. Tenpas22 May 2007 | International Journal for Computational Methods in Engineering Science and Mechanics, Vol. 8, No. 4The k-Version of Finite Element Method for Initial Value Problems: Mathematical and Computational FrameworkK. S. Surana, J. N. Reddy and S. Allu10 April 2007 | International Journal for Computational Methods in Engineering Science and Mechanics, Vol. 8, No. 3k-version of finite element method in gas dynamics: higher-order global differentiability numerical solutionsK. S. Surana, S. Allu, P. W. Tenpas and J. N. Reddy1 January 2007 | International Journal for Numerical Methods in Engineering, Vol. 69, No. 6k-Version of finite element method in 2-D polymer flows: Oldroyd-B constitutive modelK. S. Surana, A. Mohammed, J. N. Reddy and P. W. TenPas1 January 2006 | International Journal for Numerical Methods in Fluids, Vol. 52, No. 2h, p, k Least Squares Finite Element Processes for 1-D Helmholtz EquationK. S. Surana, P. Gupta, P. W. Tenpas and J. N. Reddy23 February 2007 | International Journal for Computational Methods in Engineering Science and Mechanics, Vol. 7, No. 4The k -Version Finite Element Method for Singular Boundary-Value Problems with Application to Linear Fracture MechanicsK. S. Surana, A. Rajwani and J. N. Reddy23 February 2007 | International Journal for Computational Methods in Engineering Science and Mechanics, Vol. 7, No. 3Least-squares finite element models of two-dimensional compressible flowsJ.P Pontaza, Xu Diao, J.N Reddy and K.S Surana1 Mar 2004 | Finite Elements in Analysis and Design, Vol. 40, No. 5-6 Recommended Vol. 04, No. 04 Metrics History Received 16 April 2002 Revised 14 December 2002 PDF download" @default.
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