# Modeling and Simulation Group Meeting

#### MFEM: Accelerating Efficient Solution of PDEs at Exascale

**Speaker:**
Tzanio Kolev, CASC, Livermore National Lab

**Location:**
Warren Weaver Hall 202

**Date:**
Thursday, October 31, 2024, 12:30 p.m.

**Synopsis:**

Modern GPU-based exascale architectures require rethinking of the numerical algorithms used in large-scale PDE-based applications. These architectures favor algorithms, such as high-order finite elements, that expose fine-grain parallelism and maximize the ratio of floating-point operations to energy intensive data movement.

Our approach to efficient operator evaluation is based on a "matrix-free" representation of the finite element operator, that factors a bilinear form into a series of sparse and dense components corresponding to the parallelism, mesh topology, basis, geometry, and pointwise physics in the problem. The operator decomposition exposes several layers of parallelism, enables the use of batched dgemss and tensor contractions, and only requires quadrature point values to be assembled for computing the action. This "partial assembly" formulation is a natural fit for modern HPC hardware because it results both in less (nearly optimal) computation and less (optimal) data movement compared to assembling a global sparse matrix, therefore increasing performance and reducing time to solution.

In this talk we present an overview of MFEM (https://mfem.org), a scalable library for high-order finite element discretization of PDEs on general unstructured grids that employs partial assembly and matrix-free algorithms to power a wide variety of HPC applications. In addition to discussing MFEM's capabilities and algorithms, we also report on some of the work in related projects, including high-order ALE compressible hydrodynamics in LLNL's BLAST code; GPU benchmarks from the Center for Efficient Exascale Discretizations in the US Exascale Computing Project; and scalable unstructured adaptive mesh refinement for the Energy Earthshot project "Ka mana o ka lā: Modeling our energy future" with the University of Hawaii.