Local bounded cochain projection

We construct projections from the space of differential k-forms which belong to L2 and whose exterior derivative also belongs to L2, to finite dimensional subspaces of piecewise polynomial differential forms defined on a simplicial mesh. These projections have the properties that they commute with the exterior derivative and are bounded independent of the mesh size. Unlike some other recent work in this direction, the projections are also locally defined in the sense that they are defined by local operators on overlapping macroelements, in the spirit of the Clement interpolant.

[1]  L. D. Marini,et al.  Two families of mixed finite elements for second order elliptic problems , 1985 .

[2]  Leszek Demkowicz,et al.  H1, H(curl) and H(div)-conforming projection-based interpolation in three dimensionsQuasi-optimal p-interpolation estimates , 2005 .

[3]  P. Clément.  Approximation by finite element functions using local regularization , 1975 .

[4]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[5]  H. Whitney.  Geometric Integration Theory , 1957 .

[6]  Snorre H. Christiansen,et al.  Stability of Hodge decompositions in finite element spaces of differential forms in arbitrary dimension , 2007, Numerische Mathematik.

[7]  Loring W. Tu,et al.  Differential forms in algebraic topology , 1982, Graduate texts in mathematics.

[8]  J. Nédélec.  Mixed finite elements in ℝ3 , 1980 .

[9]  Snorre H. Christiansen,et al.  Smoothed projections in finite element exterior calculus , 2007, Math. Comput..

[10]  Leszek Demkowicz,et al.  Polynomial Exact Sequences and Projection-Based Interpolation with Application to Maxwell Equations , 2008 .

[11]  D. Arnold,et al.  Finite element exterior calculus: From hodge theory to numerical stability , 2009, 0906.4325.

[12]  Leszek Demkowicz,et al.  Projection-based interpolation and automatic hp-adaptivity for finite element discretizations of elliptic and maxwell problems , 2007 .

[13]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[14]  F. Brezzi.  On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers , 1974 .

[15]  Leszek F. Demkowicz,et al.  p Interpolation Error Estimates for Edge Finite Elements of Variable Order in Two Dimensions , 2003, SIAM J. Numer. Anal..

[16]  D. Arnold,et al.  Finite element exterior calculus, homological techniques, and applications , 2006, Acta Numerica.