相关论文: Machine Learning based refinement strategies for p…
In this paper, we propose a systematic approach for accelerating finite element-type methods by machine learning for the numerical solution of partial differential equations (PDEs). The main idea is to use a neural network to learn the…
We present a novel high-order accurate nodal discontinuous Galerkin (DG) method for solving nonlinear hyperbolic systems of partial differential equations (PDEs) on fully unstructured three-dimensional polyhedral meshes. A mesh generator is…
The realization of a standard Adaptive Finite Element Method (AFEM) preserves the mesh conformity by performing a completion step in the refinement loop: in addition to elements marked for refinement due to their contribution to the global…
Algebraic multigrid (AMG) methods are among the most efficient solvers for linear systems of equations and they are widely used for the solution of problems stemming from the discretization of Partial Differential Equations (PDEs). The most…
Over the past few years, neural network methods have evolved in various directions for approximating partial differential equations (PDEs). A promising new development is the integration of neural networks with classical numerical…
The proximal Galerkin finite element method is a high-order, low-iteration complexity, nonlinear numerical method that preserves the geometric and algebraic structure of point-wise bound constraints in infinite-dimensional function spaces.…
We introduce an $hp$-version symmetric interior penalty discontinuous Galerkin finite element method (DGFEM) for the numerical approximation of the biharmonic equation on general computational meshes consisting of polygonal/polyhedral…
Deep embedding learning becomes more attractive for discriminative feature learning, but many methods still require hard-class mining, which is computationally complex and performance-sensitive. To this end, we propose Adaptive Large Margin…
We introduce a new numerical method for solving time-harmonic Maxwell's equations via the modified weak Galerkin technique. The inter-element functions of the weak Galerkin finite elements are replaced by the average of the two…
The finite element method, finite difference method, finite volume method and spectral method have achieved great success in solving partial differential equations. However, the high accuracy of traditional numerical methods is at the cost…
This paper interprets the stabilized finite element method via residual minimization as a variational multiscale method. We approximate the solution to the partial differential equations using two discrete spaces that we build on a…
The present work develops hybrid multigrid methods for high-order discontinuous Galerkin discretizations of elliptic problems. Fast matrix-free operator evaluation on tensor product elements is used to devise a computationally efficient PDE…
We explain four variants of an adaptive finite element method with cubic splines and compare their performance in simple elliptic model problems. The methods in comparison are Truncated Hierarchical B-splines with two different refinement…
The aim of this work is to introduce and analyze a finite element discontinuous Galerkin method on polygonal meshes for the numerical discretization of acoustic waves propagation through poroelastic materials. Wave propagation is modeled by…
The lowest-order Neural Approximated Virtual Element Method on polygonal elements is proposed here. This method employs a neural network to locally approximate the Virtual Element basis functions, thereby eliminating issues concerning…
We extend the semi-Lagrangian discontinuous Galerkin (SLDG) method of Einkemmer to velocity grids with adaptive mesh refinement (AMR) and to three-dimensional velocity space. The original SLDG formulation assumes uniform cell widths, which…
In this paper, we present a novel nodal integration scheme for meshfree Galerkin methods that draws on the mathematical framework of the virtual element method. We adopt linear maximum-entropy basis functions for the discretization of field…
$H^1$-conforming Galerkin methods on polygonal meshes such as VEM, BEM-FEM and Trefftz-FEM employ local finite element functions that are implicitly defined as solutions of Poisson problems having polynomial source and boundary data.…
The subject of this work is an adaptive stochastic Galerkin finite element method for parametric or random elliptic partial differential equations, which generates sparse product polynomial expansions with respect to the parametric…
This dissertation is devoted to provide advanced nonconvex nonsmooth variational models of (Magnetic Resonance Image) MRI reconstruction, efficient learnable image reconstruction algorithms and parameter training algorithms that improve the…