Główne pojęcia
This work presents efficient matrix-free algorithms for evaluating the operator action in unfitted finite element discretizations, enabling high-performance computations with high-order polynomial spaces.
Streszczenie
The key highlights and insights of this content are:
Unfitted finite element methods, such as CutFEM, have traditionally been implemented using sparse matrix-based approaches, which suffer from increasing complexity per degree of freedom as the polynomial degree increases.
To address this challenge, the authors propose a matrix-free approach that evaluates the operator action by looping over cells and faces, locally computing the cell, face, and interface integrals, including the contributions from cut cells and stabilization terms.
The main technical difficulty lies in the efficient numerical evaluation of terms in the weak form with unstructured quadrature points arising from the unfitted discretization in cut cells.
The authors present design choices and performance optimizations for tensor-product elements, including the use of sum-factorization techniques for structured quadrature and specialized algorithms for unstructured quadrature.
The performance of the proposed matrix-free algorithms is demonstrated through benchmarks and application examples, showing a speedup of more than one order of magnitude compared to sparse matrix-vector products for a discontinuous Galerkin discretization with polynomial degree three.
The authors develop performance models to quantify the performance properties of the matrix-free approach over a wide range of polynomial degrees, highlighting the benefits of high-order unfitted methods in terms of error vs. computational cost.
Statystyki
The following sentences contain key metrics or important figures used to support the author's key logics:
The authors demonstrate a speedup of more than one order of magnitude for the operator evaluation of a discontinuous Galerkin discretization with polynomial degree three compared to a sparse matrix-vector product.
The authors develop performance models to quantify the performance properties of the matrix-free approach over a wide range of polynomial degrees.