# A High-Order Lower-Triangular Pseudo-Mass Matrix for Explicit Time Advancement of hp Triangular Finite Element Methods

## TL;DR

This paper introduces a high-order lower-triangular pseudo-mass matrix for explicit time advancement in hp triangular finite element methods, enabling efficient computations without losing spatial accuracy.

## Contribution

It proposes a novel pseudo-mass matrix approach for triangular elements that maintains accuracy while improving computational efficiency in explicit time-stepping.

## Key findings

- Pseudo-mass matrix enables efficient explicit time-stepping.
- Maintains exact projection of functions in polynomial spaces.
- Applicable to high-order triangular finite element methods.

## Abstract

Explicit time advancement for continuous finite elements requires the inversion of a global mass matrix. For spectral element simulations on quadrilaterals and hexahedra, there is an accurate approximate mass matrix which is diagonal, making it computationally efficient for explicit simulations. In this article it is shown that for the standard space of polynomials used with triangular elements, denoted $\mathcal{T}(p)$ where $p$ is the degree of the space, there is no diagonal approximate mass matrix that permits accurate solutions. Accuracy is defined as giving an exact projection of functions in $\mathcal{T}(p-1)$. In light of this, a lower-triangular pseudo-mass matrix method is introduced and demonstrated for the space $\mathcal{T}(3)$. The pseudo-mass matrix and accompanying high-order basis allow for computationally efficient time-stepping techniques without sacrificing the accuracy of the spatial approximation for unstructured triangular meshes.