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This paper introduces JAX-FEM, an open-source differentiable finite element method (FEM) library. Constructed on top of Google JAX, a rising machine learning library focusing on high-performance numerical computing, JAX-FEM is implemented…

Mathematical Software · Computer Science 2023-06-28 Tianju Xue , Shuheng Liao , Zhengtao Gan , Chanwook Park , Xiaoyu Xie , Wing Kam Liu , Jian Cao

The eXtended Finite Element Method (XFEM) is used to solve interface problems with an unfitted mesh. We present an implementation of the XFEM in the FEM-library deal.II. The main parts of the implementation are (i) the appropriate…

Numerical Analysis · Mathematics 2015-07-16 Thomas Carraro , Sven Wetterauer

We study the discretisation of a uniaxial (rank-one) reduction of the Oldroyd-B model for dilute polymer solutions, in which the conformation tensor is represented as $\sig = \vec b \otimes \vec b$. Building on structural analogies with…

Numerical Analysis · Mathematics 2025-11-26 Ben S. Ashby , Gabriel R. Barrenechea , Alex Lukyanov , Tristan Pryer , Alex Trenam

Unfitted finite element methods, like CutFEM, have traditionally been implemented in a matrix-based fashion, where a sparse matrix is assembled and later applied to vectors while solving the resulting linear system. With the goal of…

Numerical Analysis · Mathematics 2024-04-15 Maximilian Bergbauer , Peter Munch , Wolfgang A. Wall , Martin Kronbichler

We extend the shifted boundary method (SBM) to the simulation of incompressible fluid flow using immersed octree meshes. Previous work on SBM for fluid flow primarily utilized two- or three-dimensional unstructured tetrahedral grids.…

We present an indirect higher order boundary element method utilising NURBS mappings for exact geometry representation and an interpolation-based fast multipole method for compression and reduction of computational complexity, to counteract…

Numerical Analysis · Mathematics 2018-02-14 Jürgen Dölz , Helmut Harbrecht , Stefan Kurz , Sebastian Schöps , Felix Wolf

We present the application of simultaneous diagonalization and minimum energy (SDME) high-order finite element modal bases for simulation of transient non-linear elastodynamic problem, including impact cases with neo-hookean hyperelastic…

Numerical Analysis · Mathematics 2021-04-29 A. P. C. Dias , J. L. Suzuki , G. L. Valente , M. L. Bittencourt

We propose, analyze mathematically, and study numerically a novel approach for the finite element approximation of the spectrum of second-order elliptic operators. The main idea is to reduce the stiffness of the problem by subtracting a…

Numerical Analysis · Mathematics 2021-07-09 Quanling Deng , Alexandre Ern

We consider numerical simulation of the isotropic elastic wave equations arising from seismic applications with non-trivial land topography. The more flexible finite element method is applied to the shallow region of the simulation domain…

Numerical Analysis · Mathematics 2018-12-26 Longfei Gao , David Keyes

Damping of structures and systems is often dominated by frictional dissipation in connections, the prediction of which remains a longstanding scientific challenge. Previous studies have shown that the actual topography of contact interfaces…

Computational Engineering, Finance, and Science · Computer Science 2026-03-30 Hendrik D. Linder , David A. Najera-Flores , Robert J. Kuether , Malte Krack

Approximating partial differential equations for extensive industrial and scientific applications requires leveraging the power of modern high-performance computing. In large-scale parallel computations, the geometrical discretisation…

Numerical Analysis · Mathematics 2024-12-30 Pere A. Martorell , Santiago Badia

We propose, analyze, and test new robust iterative solvers for systems of linear algebraic equations arising from the space-time finite element discretization of reduced optimality systems defining the approximate solution of hyperbolic…

Numerical Analysis · Mathematics 2024-04-08 Ulrich Langer , Richard Löscher , Olaf Steinbach , Huidong Yang

In this work we formally derive and prove the correctness of the algorithms and data structures in a parallel, distributed-memory, generic finite element framework that supports h-adaptivity on computational domains represented as…

Mathematical Software · Computer Science 2021-09-30 Santiago Badia , Alberto F. Martín , Eric Neiva , Francesc Verdugo

Solving multiscale diffusion problems is often computationally expensive due to the spatial and temporal discretization challenges arising from high-contrast coefficients. To address this issue, a partially explicit temporal splitting…

Numerical Analysis · Mathematics 2026-02-26 Yating Wang , Zhengya Yang , Wing Tat Leung

We propose a hybrid method, the Neural Enrichment Finite Element Method (NEFEM), designed for problems involving strong oscillations or interface problems with weak discontinuities. This method is based on the stable generalized finite…

Numerical Analysis · Mathematics 2026-05-12 Shihan Guo , Thomas Richter

Modern accelerators use hierarchical parallel programming models that enable massive multithreading within a processing element (PE), with multiple PEs per device driven by traditional processes. Batching is a technique for exposing…

Plasma Physics · Physics 2024-09-26 Mark F. Adams , Peng Wang , Jacob Merson , Kevin Huck , Matthew G. Knepley

An extremely fast time-harmonic finite element solver developed for the transmission analysis of photonic crystals was applied to mask simulation problems. The applicability was proven by examining a set of typical problems and by a…

Optics · Physics 2009-05-28 S. Burger , R. Köhle , L. Zschiedrich , W. Gao , F. Schmidt , R. März , C. Nölscher

Many problems in physics are inherently of multi-scale nature. The issues of MHD turbulence or magnetic reconnection, namely in the hot and sparse, almost collision-less astrophysical plasmas, can stand as clear examples. The Finite Element…

Computational Physics · Physics 2012-06-14 Jan Skala , Miroslav Barta

In this paper, we present a new immersed finite element scheme for solving elliptic interface problems on unfitted meshes by combining the skeletal finite element method (FEM) with the standard FEM. The skeletal FEM is used for the…

Numerical Analysis · Mathematics 2025-09-17 Lin Yang , Qilong Zhai

We present a cut finite element method for the heat equation on two overlapping meshes: a stationary background mesh and an overlapping mesh that moves around inside/"on top" of it. Here the overlapping mesh is prescribed a simple…

Numerical Analysis · Mathematics 2023-03-02 Mats G. Larson , Anders Logg , Carl Lundholm