This book describes the theoretical and computational aspects of the mimetic finite
difference method for a wide class of multidimensional elliptic problems, which includes …

This is an elementary introduction to the Hodge Laplacian on a graph, a higher-order
generalization of the graph Laplacian. We will discuss basic properties including …

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 …

The mimetic finite difference (MFD) method mimics fundamental properties of mathematical
and physical systems including conservation laws, symmetry and positivity of solutions …

The purpose of this paper is to survey some recent advances in variational integrators for
both finite dimensional mechanical systems as well as continuum mechanics. These …

Motivated by the vast success of deep convolutional networks, there is a great interest in
generalizing convolutions to non-Euclidean manifolds. A major complication in comparison …

The field of discrete calculus, also known as" discrete exterior calculus", focuses on finding a
proper set of definitions and differential operators that make it possible to operate the …

We describe a discrete Laplacian suitable for any triangle mesh, including those that are
nonmanifold or nonorientable (with or without boundary). Our Laplacian is a robust drop‐in …

Direction fields and vector fields play an increasingly important role in computer graphics
and geometry processing. The synthesis of directional fields on surfaces, or other spatial …

Constrained by the limitations of learning toolkits engineered for other applications, such as
those in image processing, many mesh-based learning algorithms employ data flows that …