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Double complexes and local cochain projections.
Richard S Falk1, Ragnar Winther2
1Department of Mathematics, Rutgers University Piscataway, New Jersey, 08854.
Numerical Methods for Partial Differential Equations
|April 28, 2015
Summary
Researchers developed new local bounded cochain projections for finite element exterior calculus. This advances stability analysis by overcoming the nonlocality of previous methods.
Area of Science:
- Mathematical analysis
- Computational mathematics
- Topology
Background:
- Bounded cochain projections are crucial for stability analysis in finite element exterior calculus.
- Current methods combine smoothing operators with canonical projections, resulting in nonlocal operators.
- Nonlocality in these projections is an undesirable property hindering certain applications.
Purpose of the Study:
- To present an alternative construction of bounded cochain projections.
- To develop bounded cochain projections that are local, unlike existing methods.
- To enhance the applicability of finite element exterior calculus in stability analysis.
Main Methods:
- Utilizing a double complex structure, analogous to the Čech-de Rham double complex.
- Developing a novel approach to construct bounded cochain projections.
- Ensuring the constructed projections commute with the exterior derivative and are bounded in Sobolev spaces.
Main Results:
- A new construction for bounded cochain projections has been established.
- These newly constructed projections exhibit locality, a significant improvement over prior methods.
- The double complex structure provides a powerful framework for this construction.
Conclusions:
- The development of local bounded cochain projections offers a significant advancement.
- This new construction is expected to improve stability analysis in finite element exterior calculus.
- The use of double complex structures opens new avenues for theoretical and computational advancements.
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