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This article considers the extension of two-grid $hp$-version discontinuous Galerkin finite element methods for the numerical approximation of second-order quasilinear elliptic boundary value problems of monotone type to the case when…
This work explores the application of the fast assembly and formation strategy from [8, 17] to trimmed bi-variate parameter spaces. Two concepts for the treatment of basis functions cut by the trimming curve are investigated: one employs a…
In marine offshore engineering, cost-efficient simulation of unsteady water waves and their nonlinear interaction with bodies are important to address a broad range of engineering applications at increasing fidelity and scale. We consider a…
We present and analyze a new embedded--hybridized discontinuous Galerkin finite element method for the Stokes problem. The method has the attractive properties of full hybridized methods, namely an $H({\rm div})$-conforming velocity field,…
Rapid growth in scientific data and a widening gap between computational speed and I/O bandwidth makes it increasingly infeasible to store and share all data produced by scientific simulations. Instead, we need methods for reducing data…
We provide both a general framework for discretizing de Rham sequences of differential forms of high regularity, and some examples of finite element spaces that fit in the framework. The general framework is an extension of the previously…
We introduce a novel kernel learning framework toward efficiently solving nonlinear partial differential equations (PDEs). In contrast to the state-of-the-art kernel solver that embeds differential operators within kernels, posing…
This note describes the full approximation storage (FAS) multigrid scheme for an easy one-dimensional nonlinear boundary value problem. The problem is discretized by a simple finite element (FE) scheme. We apply both FAS V-cycles and…
The goal of this primer is to provide a relatively short exposition of the basics of multigrid methods, simplified by focusing on fundamental concepts in a variational setting. This is done by way of a quadratic energy minimization…
Motivated by the need for efficient and accurate simulation of the dynamics of the polar ice sheets, we design high-order finite element discretizations and scalable solvers for the solution of nonlinear incompressible Stokes equations. We…
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…
In this paper, we propose an efficient quadratic interpolation formula utilizing solution gradients computed and stored at nodes and demonstrate its application to a third-order cell-centered finite-volume discretization on tetrahedral…
We advertise the use of tetrahedral grids constructed via the longest edge bisection algorithm for rendering volumetric data with path tracing. The key benefits of such grids is two-fold. First, they provide a highly adaptive…
In this work, we study the accuracy and efficiency of hierarchical matrix ($\mathcal{H}$-matrix) based fast methods for solving dense linear systems arising from the discretization of the 3D elastodynamic Green's tensors. It is well known…
The aim of this paper is to find an optimal matching between manifold-valued curves, and thereby adequately compare their shapes, seen as equivalent classes with respect to the action of reparameterization. Using a canonical decomposition…
We leverage the proximal Galerkin algorithm (Keith and Surowiec, Foundations of Computational Mathematics, 2024, DOI: 10.1007/s10208-024-09681-8), a recently introduced mesh-independent algorithm, to obtain a high-order finite element…
In this paper, we present a finite element method (FEM) framework enhanced by an operator-adapted wavelet decomposition algorithm designed for the efficient analysis of multiscale electromagnetic problems. Usual adaptive FEM approaches,…
Second order accurate Cartesian grid methods have been well developed for interface problems in the literature. However, it is challenging to develop third or higher order accurate methods for problems with curved interfaces and internal…
Superpixels have long been used in image simplification to enable more efficient data processing and storage. However, despite their computational potential, their irregular spatial distribution has often forced deep learning approaches to…
In some cases, computational benefit can be gained by exploring the hyper parameter space using a deterministic set of grid points instead of a Markov chain. We view this as a numerical integration problem and make three unique…