Related papers: General Field Evaluation in High-Order Meshes on G…
We introduce a finite element construction for use on the class of convex, planar polygons and show it obtains a quadratic error convergence estimate. On a convex n-gon satisfying simple geometric criteria, our construction produces 2n…
We present a velocity-based moving mesh virtual element method for the numerical solution of PDEs involving moving boundaries. The virtual element method is used for computing both the mesh velocity and a conservative Arbitrary…
Mesh processing pipelines are mature, but adapting them to newer non-mesh surface representations -- which enable fast rendering with compact file size -- requires costly meshing or transmitting bulky meshes, negating their core benefits…
The purpose of this research is to describe an efficient iterative method suitable for obtaining high accuracy solutions to high frequency time-harmonic scattering problems. The method allows for both refinement of local polynomial degree…
This paper introduces a framework designed to accurately predict piecewise linear mappings of arbitrary meshes via a neural network, enabling training and evaluating over heterogeneous collections of meshes that do not share a…
In this paper, we study a generalized finite element method for solving second-order elliptic partial differential equations with rough coefficients. The method uses local approximation spaces computed by solving eigenvalue problems on…
Integral equation methods for the solution of partial differential equations, when coupled with suitable fast algorithms, yield geometrically flexible, asymptotically optimal and well-conditioned schemes in either interior or exterior…
We present a matrix-free multigrid method for high-order discontinuous Galerkin (DG) finite element methods with GPU acceleration. A performance analysis is conducted, comparing various data and compute layouts. Smoother implementations are…
Mesh-free numerical methods offer flexibility in discretising complex geometries, showing potential where mesh-based methods struggle. While high-order approximations can be obtained via consistency correction using linear systems, they…
This paper presents a high-order method for solving an interface problem for the Poisson equation on embedded meshes through a coupled finite element and integral equation approach. The method is capable of handling homogeneous or…
High-order spatial discretisations and full discretisations of parabolic partial differential equations on evolving surfaces are studied. We prove convergence of the high-order evolving surface finite element method, by showing high-order…
We introduce generalised finite difference methods for solving fully nonlinear elliptic partial differential equations. Methods are based on piecewise Cartesian meshes augmented by additional points along the boundary. This allows for…
We present an algorithmic framework for matrix-free evaluation of discontinuous Galerkin finite element operators based on sum factorization on quadrilateral and hexahedral meshes. We identify a set of kernels for fast quadrature on cells…
The finite element method (FEM) has several computational steps to numerically solve a particular problem, to which many efforts have been directed to accelerate the solution stage of the linear system of equations. However, the finite…
Partial differential equations (PDEs) have become an essential tool for modeling complex physical systems. Such equations are typically solved numerically via mesh-based methods, such as finite element methods, with solutions over the…
Large-scale parallel numerical simulations are essential for a wide range of engineering problems that involve complex, coupled physical processes interacting across a broad range of spatial and temporal scales. The data structures involved…
High-quality novel view synthesis for large-scale scenes presents a challenging dilemma in 3D computer vision. Existing methods typically partition large scenes into multiple regions, reconstruct a 3D representation using Gaussian splatting…
Agglomeration-based strategies are important both within adaptive refinement algorithms and to construct scalable multilevel algebraic solvers. In order to automatically perform agglomeration of polygonal grids, we propose the use of…
In recent years, a number of finite element methods have been formulated for the solution of partial differential equations on complex geometries based on non-matching or overlapping meshes. Examples of such methods include the fictitious…
A simple MATLAB implementation of hierarchical shape functions on 2D rectangles is explained and available for download. Global shape functions are ordered for a given polynomial degree according to the indices of the nodes, edges, or…