scholarly journals Local L2-bounded commuting projections in FEEC

2021 ◽  
Author(s):  
Douglas Arnold ◽  
Johnny Guzmán
Keyword(s):  
Finite Element ◽  
Exterior Derivative ◽  
Finite Element Exterior Calculus ◽  
Exterior Calculus ◽  
Local Projections

We construct local projections into canonical finite element spaces that appear in the finite element exterior calculus. These projections are bounded in L2 and commute with the exterior derivative.

Mimetic Interpolation of Vector Fields on Arakawa C/D Grids

2019 ◽  
Vol 147 (1) ◽  
pp. 3-16
Author(s):  
Alexander Pletzer ◽  
Wolfgang Hayek
Keyword(s):  
Finite Element ◽  
Vector Fields ◽  
Total Flux ◽  
Machine Accuracy ◽  
Weighted Interpolation ◽  
Discretization Methods ◽  
Finite Element Exterior Calculus ◽  
Interpolation Methods ◽  
Discrete Exterior Calculus ◽  
Exterior Calculus

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.

scholarly journals Finite element exterior calculus, homological techniques, and applications

Acta Numerica ◽  
2006 ◽  
Vol 15 ◽  
pp. 1-155 ◽  
Author(s):  
Douglas N. Arnold ◽  
Richard S. Falk ◽  
Ragnar Winther
Keyword(s):  
Finite Element ◽  
Algebraic Topology ◽  
Eigenvalue Problems ◽  
Differential Forms ◽  
Element Discretization ◽  
Hodge Laplacian ◽  
Finite Element Exterior Calculus ◽  
Exterior Calculus ◽  
Elliptic Eigenvalue Problems ◽  
Elliptic Complex

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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