Talk by Sai Prabhav
Posted on: 05 Aug 2026
Title: Matrix-Free Discontinuous Galerkin Methods for Conservation Laws in deal.II: Sum Factorization, SIMD, and the Case Against Matrices
Speaker: Sai Prabhav, BMath, ISI Bangalore
Date: 9 Aug 2026
Time: 2 PM IST
Abstract: Discontinuous Galerkin (DG) discretizations of hyperbolic conservation laws produce semi-discrete systems M dU/dt = L(U) whose time evolution is dominated by repeated evaluation of the spatial operator L(). The standard DG approach assembles dense local element matrices cell-by-cell through matrix-matrix products. Matrix-free evaluation replaces these with matrix-vector operations via sum factorization. Because the basis is a product of one-dimensional polynomials the entire operator is written as a sequence of tensor-product contractions.
Per degree of freedom, the work of matrix-free operator evaluation grows as O(k) in the polynomial degree k, versus O(k^d) for matrix-based assembly. A further ingredient is SIMD vectorization: deal.II batches cells into SIMD lanes, so the same flux function evaluates four or more cells simultaneously.