What should you expect in intercollegiate classes?  What can you do to get the most out of them?  In this session, experienced class tutors will share their thoughts, including advice about online classes. 

All undergraduate and masters students welcome, especially Part B and MSc students attending intercollegiate classes. 

We'll discuss what mathematicians are looking for in written solutions.  How can you set out your ideas clearly, and what are the standard mathematical conventions?

This session is likely to be most relevant for first-year undergraduates, but all are welcome.

We use energy estimates to derive new bounds on the eigenvalues of a generic form of double saddle-point matrices, with and without regularization terms. Results related to inertia and algebraic multiplicity of eigenvalues are also presented. The analysis includes eigenvalue bounds for preconditioned matrices based on block-diagonal Schur complement-based preconditioners, and it is shown that in this case the eigenvalues are clustered within a few intervals bounded away from zero. The analytical observations are linked to a few multiphysics problems of interest. This is joint work with Susanne Bradley.

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A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact @email.

Many PDE models encode fundamental physical, geometric and topological structures. These structures may be lost in discretisations, and preserving them on the discrete level is crucial for the stability and efficiency of numerical methods. The finite element exterior calculus (FEEC) is a framework for constructing and analysing structure-preserving numerical methods for PDEs with ideas from topology, homological algebra and the Hodge theory. 

In this seminar, we present the theory and applications of FEEC. This includes analytic results (Hodge decomposition, regular potentials, compactness etc.), Hodge-Laplacian problems and their structure-preserving finite element discretisation, and applications in electromagnetism, fluid and solid mechanics. Knowledge on geometry and topology is not required as prerequisites.

References:

1. Arnold, D.N.: Finite Element Exterior Calculus. SIAM (2018) 

2. Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numerica 15, 1 (2006) 

3. Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus: from Hodge theory to numerical stability. Bulletin of the American Mathematical Society 47(2), 281–354 (2010) 

4. Arnold, D.N., Hu, K.: Complexes from complexes. Foundations of Computational Mathematics (2021)

Many PDE models encode fundamental physical, geometric and topological structures. These structures may be lost in discretisations, and preserving them on the discrete level is crucial for the stability and efficiency of numerical methods. The finite element exterior calculus (FEEC) is a framework for constructing and analysing structure-preserving numerical methods for PDEs with ideas from topology, homological algebra and the Hodge theory. 

In this seminar, we present the theory and applications of FEEC. This includes analytic results (Hodge decomposition, regular potentials, compactness etc.), Hodge-Laplacian problems and their structure-preserving finite element discretisation, and applications in electromagnetism, fluid and solid mechanics. Knowledge on geometry and topology is not required as prerequisites.

References:

1. Arnold, D.N.: Finite Element Exterior Calculus. SIAM (2018) 

2. Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numerica 15, 1 (2006) 

3. Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus: from Hodge theory to numerical stability. Bulletin of the American Mathematical Society 47(2), 281–354 (2010) 

4. Arnold, D.N., Hu, K.: Complexes from complexes. Foundations of Computational Mathematics (2021)

Many PDE models encode fundamental physical, geometric and topological structures. These structures may be lost in discretisations, and preserving them on the discrete level is crucial for the stability and efficiency of numerical methods. The finite element exterior calculus (FEEC) is a framework for constructing and analysing structure-preserving numerical methods for PDEs with ideas from topology, homological algebra and the Hodge theory. 

In this seminar, we present the theory and applications of FEEC. This includes analytic results (Hodge decomposition, regular potentials, compactness etc.), Hodge-Laplacian problems and their structure-preserving finite element discretisation, and applications in electromagnetism, fluid and solid mechanics. Knowledge on geometry and topology is not required as prerequisites.

References:

1. Arnold, D.N.: Finite Element Exterior Calculus. SIAM (2018) 

2. Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numerica 15, 1 (2006) 

3. Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus: from Hodge theory to numerical stability. Bulletin of the American Mathematical Society 47(2), 281–354 (2010) 

4. Arnold, D.N., Hu, K.: Complexes from complexes. Foundations of Computational Mathematics (2021)

Many PDE models encode fundamental physical, geometric and topological structures. These structures may be lost in discretisations, and preserving them on the discrete level is crucial for the stability and efficiency of numerical methods. The finite element exterior calculus (FEEC) is a framework for constructing and analysing structure-preserving numerical methods for PDEs with ideas from topology, homological algebra and the Hodge theory. 

In this seminar, we present the theory and applications of FEEC. This includes analytic results (Hodge decomposition, regular potentials, compactness etc.), Hodge-Laplacian problems and their structure-preserving finite element discretisation, and applications in electromagnetism, fluid and solid mechanics. Knowledge on geometry and topology is not required as prerequisites.

References:

1. Arnold, D.N.: Finite Element Exterior Calculus. SIAM (2018) 

2. Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numerica 15, 1 (2006) 

3. Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus: from Hodge theory to numerical stability. Bulletin of the American Mathematical Society 47(2), 281–354 (2010) 

4. Arnold, D.N., Hu, K.: Complexes from complexes. Foundations of Computational Mathematics (2021)

[REMOTE TALK]

In this talk, we will propose a new construction of the virtual fundamental classes of quasi-smooth derived schemes using the vanishing cycle complexes. This is based on the dimensional reduction theorem of cohomological Donaldson—Thomas invariants which can be regarded as a variant of the Thom isomorphism. We will also discuss a conjectural approach to construct DT4 virtual classes using the vanishing cycle complexes.

Zoom link: https://us02web.zoom.us/j/86267335498?pwd=R2hrZ1N3VGJYbWdLd0htZzA4Mm5pd…