# Approximation of tensor fields on surfaces of arbitrary topology based on local Monge parametrizations

## TL;DR

The paper introduces the Local Monge Parametrizations (LMP) method for approximating tensor fields on arbitrary surfaces, enabling efficient numerical solutions of tensor PDEs without increasing PDE order or restricting surface topology.

## Contribution

The LMP method provides a general, efficient approach for tensor approximation on complex surfaces using local Monge maps, overcoming limitations of previous methods.

## Key findings

- LMP accurately approximates vector and tensor fields on various surfaces.
- LMP successfully solves physical tensor PDEs on complex topologies.
- The method uses optimal degrees of freedom for tensor representation.

## Abstract

We introduce a new method, the Local Monge Parametrizations (LMP) method, to approximate tensor fields on general surfaces given by a collection of local parametrizations, e.g.~as in finite element or NURBS surface representations. Our goal is to use this method to solve numerically tensor-valued partial differential equations (PDE) on surfaces. Previous methods use scalar potentials to numerically describe vector fields on surfaces, at the expense of requiring higher-order derivatives of the approximated fields and limited to simply connected surfaces, or represent tangential tensor fields as tensor fields in 3D subjected to constraints, thus increasing the essential number of degrees of freedom. In contrast, the LMP method uses an optimal number of degrees of freedom to represent a tensor, is general with regards to the topology of the surface, and does not increase the order of the PDEs governing the tensor fields. The main idea is to construct maps between the element parametrizations and a local Monge parametrization around each node. We test the LMP method by approximating in a least-squares sense different vector and tensor fields on simply connected and genus-1 surfaces. Furthermore, we apply the LMP method to two physical models on surfaces, involving a tension-driven flow (vector-valued PDE) and nematic ordering (tensor-valued PDE). The LMP method thus solves the long-standing problem of the interpolation of tensors on general surfaces with an optimal number of degrees of freedom.