scholarly journals Geometric Variational Crimes: Hilbert Complexes, Finite Element Exterior Calculus, and Problems on Hypersurfaces

2012 ◽  
Vol 12 (3) ◽  
pp. 263-293 ◽  
Author(s):  
Michael Holst ◽  
Ari Stern
2019 ◽  
Vol 147 (1) ◽  
pp. 3-16
Author(s):  
Alexander Pletzer ◽  
Wolfgang Hayek

Interpolation methods for vector fields whose components are staggered on horizontal Arakawa C or D grids are presented. The interpolation methods extend bilinear and area-weighted interpolation, which are widely used in Earth sciences, to work with vector fields (essentially discretized versions of differential 1-forms and 2-forms). The interpolation methods, which conserve the total flux and enforce Stokes’ theorem to near-machine accuracy, are a natural complement to discrete exterior calculus and finite element exterior calculus discretization methods.


Acta Numerica ◽  
2006 ◽  
Vol 15 ◽  
pp. 1-155 ◽  
Author(s):  
Douglas N. Arnold ◽  
Richard S. Falk ◽  
Ragnar Winther

Finite element exterior calculus is an approach to the design and understanding of finite element discretizations for a wide variety of systems of partial differential equations. This approach brings to bear tools from differential geometry, algebraic topology, and homological algebra to develop discretizations which are compatible with the geometric, topological, and algebraic structures which underlie well-posedness of the PDE problem being solved. In the finite element exterior calculus, many finite element spaces are revealed as spaces of piecewise polynomial differential forms. These connect to each other in discrete subcomplexes of elliptic differential complexes, and are also related to the continuous elliptic complex through projections which commute with the complex differential. Applications are made to the finite element discretization of a variety of problems, including the Hodge Laplacian, Maxwell’s equations, the equations of elasticity, and elliptic eigenvalue problems, and also to preconditioners.


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