**Variational and Geometric
Aspects
of Compatible Discretizations**

**How different people
discretize**

**A sequence of linear
systems vs.
a single linear system**

**A summary of variational
settings for FEM**

**What does variational
compatibility buy you**

**What does variational
compatibility
say about the other issues?**

**Problems with identical
reduced systems**

**How to achieve geometric
compatibility?**

**Algebraic Topology Approach
1. System reduction**

**Algebraic Topology Approach
2. Inner products and dual operators**

**Algebraic Topology
Framework: Summary**

**Direct discretization of a
div-curl system**

**Direct discretization of a
div-grad system**

**What does geometric
compatibility buy you?**

**There’s only one low-order
compatible method**