Biorthogonal bases with local support and approximation properties
Authors:
Bishnu P. Lamichhane and Barbara I. Wohlmuth
Journal:
Math. Comp. 76 (2007), 233249
MSC (2000):
Primary 65N30, 65N55, 65D32
Published electronically:
October 11, 2006
MathSciNet review:
2261019
Fulltext PDF Free Access
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References 
Similar Articles 
Additional Information
Abstract: We construct locally supported basis functions which are biorthogonal to conforming nodal finite element basis functions of degree in one dimension. In contrast to earlier approaches, these basis functions have the same support as the nodal finite element basis functions and reproduce the conforming finite element space of degree . Working with GaußLobatto nodes, we find an interesting connection between biorthogonality and quadrature formulas. One important application of these newly constructed biorthogonal basis functions are twodimensional mortar finite elements. The weak continuity condition of the constrained mortar space is realized in terms of our new dual bases. As a result, local static condensation can be applied which is very attractive from the numerical point of view. Numerical results are presented for cubic mortar finite elements.
 [Ben99]
F. Ben Belgacem.
The mortar finite element method with Lagrange multipliers. Numer. Math., 84:173197, 1999.
 [Ben04]
F. Ben Belgacem.
A stabilized domain decomposition method with nonmatching grids for the stokes problem in three dimensions. SIAM J. Numer. Anal., 42:667685, 2004.
 [BMP93]
C. Bernardi, Y. Maday, and A.T. Patera.
Domain decomposition by the mortar element method. In H. Kaper et al., editor, Asymptotic and numerical methods for partial differential equations with critical parameters, pages 269286. Reidel, Dordrecht, 1993.
 [BMP94]
C. Bernardi, Y. Maday, and A.T. Patera.
A new nonconforming approach to domain decomposition: the mortar element method. In H. Brezzi et al., editor, Nonlinear partial differential equations and their applications, pages 1351. Paris, 1994.
 [Bos91]
L. Bos.
On certain configurations of points in which are unisolvent for polynomial interpolation. J. Approx. Theory, 64:271280, 1991.
 [Bru97]
L. Brutman.
Lebesgue functions for polynomial interpolationa survey. Ann. Numer. Math., 4:111127, 1997.
 [BTW00]
L. Bos, M.A. Taylor, and B.A. Wingate.
Tensor product GaussLobatto points are Fekete points for the cube. Math. Comp., 70:15431547, 2000.
 [DS99]
W. Dahmen and R.P. Stevenson.
Elementbyelement construction of wavelets satisfying stability and moment conditions. SIAM J. Numer. Anal., 37(1):319352, 1999.
 [Hes98]
J. S. Hesthaven.
From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex. SIAM J. Numer. Anal., 35:655676, 1998.
 [HT04a]
P. Hauret and P. Le Tallec.
Stabilized discontinuous mortar formulation for elastostatics and elastodynamics problems  Part I: formulation and analysis. CMAP Internal Report 553, 2004.
 [HT04b]
P. Hauret and P. Le Tallec.
A stabilized discontinuous mortar formulation for elastostatics and elastodynamics problems  Part II: discontinuous Lagrange multipliers. CMAP Internal Report 554, 2004.
 [KLPV01]
C. Kim, R.D. Lazarov, J.E. Pasciak, and P.S. Vassilevski.
Multiplier spaces for the mortar finite element method in three dimensions. SIAM J. Numer. Anal., 39:519538, 2001.
 [LSW05]
B.P. Lamichhane, R.P. Stevenson, and B.I. Wohlmuth.
Higher order mortar finite element methods in 3D with dual Lagrange multiplier bases. Numer. Math., 102:93121, 2005.
 [LW02]
B.P. Lamichhane and B.I. Wohlmuth.
Higher order dual Lagrange multiplier spaces for mortar finite element discretizations. CALCOLO, 39:219237, 2002.
 [Mar05]
L. Marcinkowski.
A mortar finite element method for fourth order problems in two dimensions with Lagrange multipliers. SIAM J. Numer. Anal., 42:19982019, 2005.
 [OW01]
P. Oswald and B.I. Wohlmuth.
On polynomial reproduction of dual FE bases. In N. Debit, M. Garbey, R. Hoppe, J. Pèriaux, D. Keyes, and Y. Kuznetsov, editors, Thirteenth International Conference on Domain Decomposition Methods, pages 8596, 2001.
 [PR04]
R. Pasquetti and F. Rapetti.
Spectral element methods on triangles and quadrilaterals: comparisons and applications. J. Comp. Phy., 198:349362, 2004.
 [Ste03]
R. Stevenson.
Locally supported, piecewise polynomial biorthogonal wavelets on nonuniform meshes. Constr. Approx., 19:477508, 2003.
 [TWV00]
M.A. Taylor, B.A. Wingate, and R.E. Vincent.
An algorithm for computing Fekete points in the triangle. SIAM J. Numer. Anal., 38:17071720, 2000.
 [Woh00]
B.I. Wohlmuth.
A mortar finite element method using dual spaces for the Lagrange multiplier. SIAM J. Numer. Anal., 38:9891012, 2000.
 [Woh01]
B.I. Wohlmuth.
Discretization Methods and Iterative Solvers Based on Domain Decomposition, volume 17 of LNCS. Springer Heidelberg, 2001.
 [WPH00]
T. Warburton, L.F. Pavarino, and J.S. Hesthaven.
A pseudospectral scheme for the incompressible NavierStokes equations using unstructured nodal elements. J. Comp. Phy., 164:121, 2000.
 [Ben99]
 F. Ben Belgacem.
The mortar finite element method with Lagrange multipliers. Numer. Math., 84:173197, 1999.
 [Ben04]
 F. Ben Belgacem.
A stabilized domain decomposition method with nonmatching grids for the stokes problem in three dimensions. SIAM J. Numer. Anal., 42:667685, 2004.
 [BMP93]
 C. Bernardi, Y. Maday, and A.T. Patera.
Domain decomposition by the mortar element method. In H. Kaper et al., editor, Asymptotic and numerical methods for partial differential equations with critical parameters, pages 269286. Reidel, Dordrecht, 1993.
 [BMP94]
 C. Bernardi, Y. Maday, and A.T. Patera.
A new nonconforming approach to domain decomposition: the mortar element method. In H. Brezzi et al., editor, Nonlinear partial differential equations and their applications, pages 1351. Paris, 1994.
 [Bos91]
 L. Bos.
On certain configurations of points in which are unisolvent for polynomial interpolation. J. Approx. Theory, 64:271280, 1991.
 [Bru97]
 L. Brutman.
Lebesgue functions for polynomial interpolationa survey. Ann. Numer. Math., 4:111127, 1997.
 [BTW00]
 L. Bos, M.A. Taylor, and B.A. Wingate.
Tensor product GaussLobatto points are Fekete points for the cube. Math. Comp., 70:15431547, 2000.
 [DS99]
 W. Dahmen and R.P. Stevenson.
Elementbyelement construction of wavelets satisfying stability and moment conditions. SIAM J. Numer. Anal., 37(1):319352, 1999.
 [Hes98]
 J. S. Hesthaven.
From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex. SIAM J. Numer. Anal., 35:655676, 1998.
 [HT04a]
 P. Hauret and P. Le Tallec.
Stabilized discontinuous mortar formulation for elastostatics and elastodynamics problems  Part I: formulation and analysis. CMAP Internal Report 553, 2004.
 [HT04b]
 P. Hauret and P. Le Tallec.
A stabilized discontinuous mortar formulation for elastostatics and elastodynamics problems  Part II: discontinuous Lagrange multipliers. CMAP Internal Report 554, 2004.
 [KLPV01]
 C. Kim, R.D. Lazarov, J.E. Pasciak, and P.S. Vassilevski.
Multiplier spaces for the mortar finite element method in three dimensions. SIAM J. Numer. Anal., 39:519538, 2001.
 [LSW05]
 B.P. Lamichhane, R.P. Stevenson, and B.I. Wohlmuth.
Higher order mortar finite element methods in 3D with dual Lagrange multiplier bases. Numer. Math., 102:93121, 2005.
 [LW02]
 B.P. Lamichhane and B.I. Wohlmuth.
Higher order dual Lagrange multiplier spaces for mortar finite element discretizations. CALCOLO, 39:219237, 2002.
 [Mar05]
 L. Marcinkowski.
A mortar finite element method for fourth order problems in two dimensions with Lagrange multipliers. SIAM J. Numer. Anal., 42:19982019, 2005.
 [OW01]
 P. Oswald and B.I. Wohlmuth.
On polynomial reproduction of dual FE bases. In N. Debit, M. Garbey, R. Hoppe, J. Pèriaux, D. Keyes, and Y. Kuznetsov, editors, Thirteenth International Conference on Domain Decomposition Methods, pages 8596, 2001.
 [PR04]
 R. Pasquetti and F. Rapetti.
Spectral element methods on triangles and quadrilaterals: comparisons and applications. J. Comp. Phy., 198:349362, 2004.
 [Ste03]
 R. Stevenson.
Locally supported, piecewise polynomial biorthogonal wavelets on nonuniform meshes. Constr. Approx., 19:477508, 2003.
 [TWV00]
 M.A. Taylor, B.A. Wingate, and R.E. Vincent.
An algorithm for computing Fekete points in the triangle. SIAM J. Numer. Anal., 38:17071720, 2000.
 [Woh00]
 B.I. Wohlmuth.
A mortar finite element method using dual spaces for the Lagrange multiplier. SIAM J. Numer. Anal., 38:9891012, 2000.
 [Woh01]
 B.I. Wohlmuth.
Discretization Methods and Iterative Solvers Based on Domain Decomposition, volume 17 of LNCS. Springer Heidelberg, 2001.
 [WPH00]
 T. Warburton, L.F. Pavarino, and J.S. Hesthaven.
A pseudospectral scheme for the incompressible NavierStokes equations using unstructured nodal elements. J. Comp. Phy., 164:121, 2000.
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Additional Information
Bishnu P. Lamichhane
Affiliation:
Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, Stuttgart, Germany
Email:
lamichhane@mathematik.unistuttgart.de
Barbara I. Wohlmuth
Affiliation:
Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, Stuttgart, Germany
Email:
wohlmuth@mathematik.unistuttgart.de
DOI:
http://dx.doi.org/10.1090/S0025571806019077
PII:
S 00255718(06)019077
Keywords:
Biorthogonal basis,
domain decomposition,
Lagrange multipliers,
reproduction property
Received by editor(s):
April 7, 2005
Received by editor(s) in revised form:
October 20, 2005
Published electronically:
October 11, 2006
Additional Notes:
This work was supported in part by the Deutsche Forschungsgemeinschaft, SFB 404, C12.
Article copyright:
© Copyright 2006 American Mathematical Society
