Doedel, Eusebius On the construction of discretizations of elliptic partial differential equations. (English) Zbl 0907.65117 J. Difference Equ. Appl. 3, No. 5-6, 389-416 (1998). This paper deals with a class of finite element (FE) collocation methods for the approximate numerical solution of elliptic partial differential equations (PDEs). The main motivation for considering these methods is their potential usefulness in the numerical study of nonlinear phenomena in PDEs by continuation and bifurcation techniques. The author considers only algorithmic aspects and discusses relatively simple equations and domains, in order to focus on the basic ideas of the discretization method and the solution procedure. For linear elliptic PDEs the methods are equivalently defined as a type of generalized finite difference methods. The finite difference formulation leads in a very natural way to the nested dissection procedure for solving the discretized equations. Some numerical results are presented. Reviewer: P.Chocholatý (Bratislava) Cited in 2 Documents MSC: 65N35 Spectral, collocation and related methods for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations 35J65 Nonlinear boundary value problems for linear elliptic equations 65N06 Finite difference methods for boundary value problems involving PDEs Keywords:numerical examples; finite element collocation methods; continuation; bifurcation; generalized finite difference methods; nested dissection procedure Software:COLSYS PDFBibTeX XMLCite \textit{E. Doedel}, J. Difference Equ. Appl. 3, No. 5--6, 389--416 (1998; Zbl 0907.65117) Full Text: DOI References: [1] Ladas G., Trans. Amer. Math. Soc. 285 pp 81– (1984) · doi:10.1090/S0002-9947-1984-0748831-8 [2] Ladas G., J. Appl. Math. Simulation 2 pp 101– (1989) [3] Ladas G., J. Math. Anal. Appl. 153 pp 276– (1990) · Zbl 0718.39002 · doi:10.1016/0022-247X(90)90278-N [4] Kocic V.L., Global Behavior of Nonlinear Difference Equations of Higher Order with Applications (1993) · Zbl 0787.39001 [5] Šilijak D.D., Nonlinear Systems (1969) [6] Šiljak D.D., IEE Proc 117 pp 2033– (1970) [7] Barnett S., SIAM Rev. 19 pp 472– (1977) · Zbl 0361.93003 · doi:10.1137/1019070 [8] Šiljak D.D., Proc. Amer. Control. Conf. Pittsburgh, PA 1 pp 193– (1989) [9] Šiljak D.D., Math. Problems Eng. 1 (1989) [10] Kharitonov V.L., Diff. Uravneniya 14 pp 2086– (1978) [11] Barmish B.R., New Tools for Robustness of Linear Systems (1994) · Zbl 1094.93517 [12] Bhattacharyya S.P., Robust Control The Parametric Approach (1995) · Zbl 0838.93008 [13] Zadeh L.A., Linear System Theory (1963) [14] Tesi A., Automatica 27 pp 147– (1991) · Zbl 0732.93062 · doi:10.1016/0005-1098(91)90013-R [15] Fruchter G.E., Automatica 27 pp 501– (1991) · Zbl 0754.93027 · doi:10.1016/0005-1098(91)90107-D [16] Lin Y.Z, J. Diff. Equa. Appl. 2 pp 301– (1996) · Zbl 0884.39006 · doi:10.1080/10236199608808064 [17] Malan, S., Milanese, M. and Taragna, M. 1996. Robust analysis and design of control systems using interval arithmetic. Proc. 13th IFAC World Congress. 1996, San Francisco, CA. pp.25–30. H · Zbl 0890.93034 [18] Liu B., J. Diff. Equa. Appl. 1 pp 307– (1995) · Zbl 0856.39017 · doi:10.1080/10236199508808029 [19] Pang P.Y.H., J. Diff. Equa. Appl. 2 pp 271– (1996) · Zbl 0884.39005 · doi:10.1080/10236199608808062 [20] Li B., J. Diff. Equa. Appl. 2 pp 389– (1996) · Zbl 0881.39007 · doi:10.1080/10236199608808073 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.