Variational and Geometric Aspects
of Compatible Discretizations

Why are we here?

Acknowledgments and Sources

How different people discretize

What do we want to know?

A sequence of linear systems vs.
a single linear system

Stability of a sequence

Variational Methods

Variational settings for FEM

Examples

No Optimization

Compatibility

Constrained Optimization

Compatibility

Slide 15

Unconstrained Optimization

A summary of variational settings for FEM

What does variational compatibility buy you

What does variational compatibility
 say about the other issues?

“Pure” Direct Discretizations

The Hodge

Problems with identical reduced systems

Matrix Form

Geometric compatibility

How to achieve geometric compatibility?

Algebraic Topology Approach
1. System reduction

Example

Algebraic Topology Approach
2. Inner products and dual operators

Examples

Properties

Algebraic Topology Framework: Summary

Direct discretization of a div-curl system

Direct discretization of a div-grad system

What does geometric compatibility buy you?

Solvability: free of charge

Variational vs. geometric

Variational and geometric

Early examples

Early examples

Can this be an accident?

Elsewhere…

CDP 1 + CDP 2 = VC

There’s only one low-order
compatible method

There are more high-order methods

Conclusions

Conclusions

Another viewpoint

Factorization (Tonti) diagrams