Madden, Niall Stynes, Martin
Published in
Calcolo

A new finite element method is presented for a general class of singularly perturbed reaction-diffusion problems -ε2Δu+bu=f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document...

Geist, Moritz Petersen, Philipp Raslan, Mones Schneider, Reinhold Kutyniok, Gitta
Published in
Journal of Scientific Computing

We perform a comprehensive numerical study of the effect of approximation-theoretical results for neural networks on practical learning problems in the context of numerical analysis. As the underlying model, we study the machine-learning-based solution of parametric partial differential equations. Here, approximation theory for fully-connected neur...

Kapl, Mario Vitrih, Vito
Published in
Advances in Computational Mathematics

The design of globally Cs-smooth (s ≥ 1) isogeometric spline spaces over multi-patch geometries with possibly extraordinary vertices, i.e. vertices with valencies different from four, is a current and challenging topic of research in the framework of isogeometric analysis. In this work, we extend the recent methods Kapl et al. Comput. Aided Geom. D...

Sheng, Qiwei Wang, Cheng
Published in
Journal of Scientific Computing

In this paper, we propose and analyze a spherical harmonic discontinuous Galerkin (SH-DG) method for solving the radiative transfer equations with vacuum boundary conditions. To incorporate vacuum boundary conditions in spherical harmonic approximations, we first embed the original domain into a larger computational area of rectangular type with an...

Harris, Isaac
Published in
Research in the Mathematical Sciences

In this paper, we consider the numerical approximation of the Steklov eigenvalue problem that arises in inverse acoustic scattering. The underlying scattering problem is for an inhomogeneous isotropic medium. These eigenvalues have been proposed to be used as a target signature since they can be recovered from the scattering data. A Galerkin method...

Zhang, Yingjuan Li, Gang Qian, Shouguo Gao, Jinmei
Published in
Computational and Applied Mathematics

This article develops a new discontinuous Galerkin (DG) method with the one-stage arbitrary derivatives in time and space approach to solve one-dimensional hyperbolic conservation laws. This method employs the differential transformation procedure instead of the Cauchy–Kowalewski procedure to recursively express the spatiotemporal expansion coeffic...

Lepe, Felipe Otárola, Enrique Quero, Daniel
Published in
Journal of Scientific Computing

We analyze, on two dimensional polygonal domains, classical low–order inf-sup stable finite element approximations of the stationary Navier–Stokes equations with singular sources. We operate under the assumptions that the continuous and discrete solutions are sufficiently small. We perform an a priori error analysis on convex domains. On Lipschitz,...

Charati, AllahBakhsh Yazdani Momeni, Hamid Cheichan, Mohammed S.
Published in
Computational and Applied Mathematics

In this work, the weak Galerkin finite element method (WG-FEM) is challenged by choosing a combination of the lowest degree of polynomial space for second-order elliptic problems. In this new scheme, we use the new stabilizer term. This scheme features piecewise-constant in each element T and piecewise-constant on ∂T\documentclass[12pt]{minimal} \u...

Liu, Huan Zheng, Xiangcheng Chen, Chuanjun Wang, Hong
Published in
Advances in Computational Mathematics

In this paper, we study the solute transport in heterogeneous media described by the time-fractional mobile/immobile advection diffusion model, where the integer and the fractional time derivatives are employed to characterize the movement of the particles in the mobile and immobile zone, respectively. We propose a fully discrete characteristic fin...

Yang, Yun-Bo Jiang, Yao-Lin Yu, Bo-Hao
Published in
Journal of Scientific Computing

This paper is concerned with unconditionally optimal error estimates of linearized leap-frog Galerkin finite element methods (FEMs) to numerically solve the d-dimensional (d=2,3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upg...