Previous |  Up |  Next


$n$-simplex; finite element method; superconvergence; strengthened Cauchy-Schwarz inequality; discrete maximum principle
Over the past fifty years, finite element methods for the approximation of solutions of partial differential equations (PDEs) have become a powerful and reliable tool. Theoretically, these methods are not restricted to PDEs formulated on physical domains up to dimension three. Although at present there does not seem to be a very high practical demand for finite element methods that use higher dimensional simplicial partitions, there are some advantages in studying the methods independent of the dimension. For instance, it provides additional insights into the structure and essence of proofs of results in one, two and three dimensions. In this survey paper we review some recent progress in this direction.
[1] B.  Achchab, S.  Achchab, O.  Axelsson, and A.  Souissi: Upper bound of the constant in strengthened C.B.S.  inequality for systems of linear partial differential equations. Numer. Algorithms 32 (2003), 185–191. DOI 10.1023/A:1024058625449 | MR 1989366
[2] B.  Achchab, O.  Axelsson, A.  Laayouni, and A.  Souissi: Strengthened Cauchy-Bunyakowski-Schwarz inequality for a three-dimensional elasticity system. Numer. Linear Algebra Appl. 8 (2001), 191–205. DOI 10.1002/1099-1506(200104/05)8:3<191::AID-NLA229>3.0.CO;2-7 | MR 1817796
[3] D. N.  Arnold, R.  Falk, and R.  Winther: Finite element exterior calculus. Acta Numer. 15 (2006), 1–135. DOI 10.1017/S0962492906210018 | MR 2269741
[4] O.  Axelsson: On multigrid methods of the two-level type. In: Multigrid Methods. Lecture Notes in Mathematics, Vol. 960, W. Hackbusch, U.  Trotenberg (eds.), Springer-Verlag, Berlin, 1982, pp. 352–367. MR 0685778 | Zbl 0505.65040
[5] O.  Axelsson, R.  Blaheta: Two simple derivations of universal bounds for the CBS  inequality constant. Appl. Math. 49 (2004), 57–72. DOI 10.1023/B:APOM.0000024520.06175.8b | MR 2032148
[6] R.  Blaheta: Nested tetrahedral grids and strengthened CBS  inequality. Numer. Linear Algebra Appl. 10 (2003), 619–637. DOI 10.1002/nla.340 | MR 2030627
[7] R.  Blaheta, S.  Margenov, and M.  Neytcheva: Uniform estimates of the constant in the strengthened CBS  inequality for anisotropic non-conforming FEM systems. Numer. Linear Algebra Appl. 11 (2004), 309–326. DOI 10.1002/nla.350 | MR 2057704
[8] D.  Braess: Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, 2nd edition. Cambridge University Press, Cambridge, 2001, pp. 309–326. MR 1827293
[9] J. H.  Brandts, S.  Korotov, and M.  Křížek: The strengthened Cauchy-Bunyakowski-Schwarz inequality for $n$-simplicial linear finite elements. In: Springer Lecture Notes in Computer Science, Vol. 3401, Springer-Verlag, Berlin, 2005, pp. 203–210.
[10] J. H.  Brandts, S.  Korotov, and M.  Křížek: Survey of discrete maximum principles for linear elliptic and parabolic problems. In: Proc. Conf. ECCOMAS  2004, P.  Neittaanmäki et al. (eds.), Univ. of Jyväskylä, 2004, pp. 1–19.
[11] J. H.  Brandts, S.  Korotov, and M.  Křížek: Dissection of the path-simplex in  $\mathbb{R}^n$ into $n$  path-subsimplices. Linear Algebra Appl. 421 (2007), 382–393. MR 2294350
[12] J. H.  Brandts, M.  Křížek: Gradient superconvergence on uniform simplicial partitions of polytopes. IMA  J.  Numer. Anal. 23 (2003), 489–505. DOI 10.1093/imanum/23.3.489 | MR 1987941
[13] S.  Brenner, L. R.  Scott: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics  15. Springer-Verlag, New York, 1994. MR 1278258
[14] P. Ciarlet: The Finite Element Method for Elliptic Problems. North Holland, Amsterdam, 1978. MR 0520174 | Zbl 0383.65058
[15] C. M.  Chen: Optimal points of stresses for tetrahedron linear element. Nat. Sci. J.  Xiangtan Univ. 3 (1980), 16–24. (Chinese)
[16] COMSOL, Multiphysics Version  3.3  (2006). Sweden,
[17] H. S. M.  Coxeter: Trisecting an orthoscheme. Comput. Math. Appl. 17 (1989), 59–71. DOI 10.1016/0898-1221(89)90148-X | MR 0994189 | Zbl 0706.51019
[18] FEMLAB version  2.2  (2002). Multiphysics in Matlab, for use with Matlab. COMSOL, Sweden,
[19] H.  Fujii: Some remarks on finite element analysis of time-dependent field problems. In: Theory Pract. Finite Elem. Struct. Anal, Univ. Tokyo Press, Tokyo, 1973, pp. 91–106. Zbl 0373.65047
[20] G.  Goodsell: Pointwise superconvergence of the gradient for the linear tetrahedral element. Numer. Methods Partial Differ. Equations 10 (1994), 651–666. DOI 10.1002/num.1690100511 | MR 1290950 | Zbl 0807.65112
[21] J.  Karátson, S.  Korotov: Discrete maximum principles for finite element solutions of nonlinear elliptic problems with mixed boundary conditions. Numer. Math. 99 (2005), 669–698. DOI 10.1007/s00211-004-0559-0 | MR 2121074
[22] M.  Křížek, Q.  Lin: On the diagonal dominance of stiffness matrices in  3D. East-West J.  Numer. Math. 3 (1995), 59–69. MR 1331484
[23] M.  Křížek, P. Neittaanmäki: On superconvergence techniques. Acta Appl. Math. 9 (1987), 175–198. DOI 10.1007/BF00047538 | MR 0900263
[24] Finite Element Methods: Superconvergence, Post-processing and A  Posteriori Estimates. Proc. Conf. Univ. of Jyväskylä, 1996. Lecture Notes in Pure and Applied Mathematics, Vol. 196. M.  Křížek, P.  Neittaanmäki, and R.  Stenberg (eds.), Marcel Dekker, New York, 1998. MR 1602809
[25] J. C.  Nédélec: Mixed finite elements in  $\mathbb{R}^3$. Numer. Math. 35 (1980), 315–341. DOI 10.1007/BF01396415
[26] J. C.  Nédélec: A new family of mixed finite elements in  $\mathbb{R}^3$. Numer. Math. 50 (1986), 57–81. DOI 10.1007/BF01389668 | MR 0864305
[27] L. A.  Oganesjan, L. A.  Ruhovets: Study of the rate of convergence of variational difference schemes for second-order elliptic equations in a two-dimensional field with a smooth boundary. Zh. Vychisl. Mat. Mat. Fiz. 9 (1969), 1102–1120. MR 0295599
[28] V.  Ruas Santos: On the strong maximum principle for some piecewise linear finite element approximate problems of non-positive type. J.  Fac. Sci., Univ. Tokyo, Sect.  IA Math. 29 (1982), 473–491. MR 0672072 | Zbl 0488.65052
[29] R. P.  Stevenson: An optimal adaptive finite element method. SIAM J.  Numer. Anal. 42 (2005), 2188–2217. DOI 10.1137/S0036142903425082 | MR 2139244 | Zbl 1081.65112
[30] P.  Tong: Exact solutions of certain problems by finite-element method. AIAA J. 7 (1969), 178–180. DOI 10.2514/3.5067
Partner of
EuDML logo