THE k-VERSION OF FINITE ELEMENT METHOD FOR SELF-ADJOINT OPERATORS IN BVP

Abstract

International Journal of Computational Engineering ScienceVol. 03, No. 02, pp. 155-218 (2002) No AccessTHE k-VERSION OF FINITE ELEMENT METHOD FOR 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/S1465876302000605Cited by:42 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail AbstractIn this paper a new mathematical and computational framework for boundary value problems described by self-adjoint differential operators is presented. In this framework, numerically computed solutions, when converged, possess the same degree of global smoothness in terms of differentiability up to any desired order as the theoretical solutions. This is accomplished using spaces Ĥk,p that contain basis functions of degree p and order k - 1 (or the order of the space k). It is shown that the order of space k is an intrinsically important independent parameter in all finite element computational processes in addition to the discretization characteristic length h and the degree of basis functions p when the theoretical solutions are analytic. Thus, in all finite element computations, all quantities of interest (e.g., quadratic functional, error or residual functional, norms and seminorms, error norms, etc.) are dependent on h, p as well as k. Therefore, for fixed h and p, convergence of the finite element process can also be investigated by changing k, hence k-convergence and thus the k-version of finite element method. With h, p, and k as three independent parameters influencing all finite element processes, we now have k, hk, pk, and hpk versions of finite element methods. The issue of minimally conforming finite element spaces is reexamined and it is demonstrated that the definition of currently believed minimally conforming space which permit weak convergence of the highest-order derivatives of the dependent variables appearing in the bilinear form is not justifiable mathematically or from physics view point. A new criterion is proposed for establishing the minimally conforming spaces which is more in agreement with the physics and mathematics of the BVP. Significant features and merits of the proposed mathematical and computational framework are presented, discussed, illustrated, and substantiated mathematically as well as numerically with the Galerkin and least-squares finite element formulations for self-adjoint boundary-value problems. FiguresReferencesRelatedDetailsCited By 42k-Version of Finite Element Method for BVPs and IVPsKaran S. Surana, Celso H. 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Sangeeta, Somenath Mukherjee and Gangan Prathap1 Jan 2006 | International Journal for Computational Methods in Engineering Science and Mechanics, Vol. 7, No. 1THE K-VERSION OF FINITE ELEMENT METHOD FOR NONLINEAR OPERATORS IN BVPK. S. SURANA, A. R. AHMADI, and J. N. REDDY10 April 2012 | International Journal of Computational Engineering Science, Vol. 05, No. 01Least-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. 03, No. 02 Metrics History PDF download