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This paper analyzes a time-stepping discontinuous Galerkin method for fractional diffusion-wave problems. This method uses piecewise constant functions in the temporal discretization and continuous piecewise linear functions in the spatial…

Numerical Analysis · Mathematics 2019-08-27 Binjie Li , Tao Wang , Xiaoping Xie

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.…

Numerical Analysis · Mathematics 2024-12-18 Brendan Keith , Thomas M. Surowiec

We investigate the well-posedness of Hasegawa-Mima equation (HME) in bounded domain with Dirichlet boundary condition and singular density under different regularity assumptions on the data. Our approach relies on the coupling of a…

Analysis of PDEs · Mathematics 2025-09-23 Franco Flandoli , Yassine Tahraoui

The discontinuous Galerkin time-stepping method has many advantageous properties for solving parabolic equations. However, it requires the solution of a large nonsymmetric system at each time-step. This work develops a fully robust and…

Numerical Analysis · Mathematics 2025-01-29 Iain Smears

We prove that the most common filtering procedure for nodal discontinuous Galerkin (DG) methods is stable. The proof exploits that the DG approximation is constructed from polynomial basis functions and that integrals are approximated with…

Numerical Analysis · Mathematics 2020-07-15 Jan Nordström , Andrew R. Winters

In this article we will investigate the large time behavior of solutions of a special class of initial/boundary value problems that involve nonlinear damped beam equations. We will show that the solution energies of global pseudo classical…

Analysis of PDEs · Mathematics 2025-11-04 David Raske

We consider a simple initial-boundary-value problem for the shallow water equations in one space dimension. We discretize the problem in space by the standard Galerkin finite element method on a quasiuniform mesh and in time by the…

Numerical Analysis · Mathematics 2018-10-26 D. c. Antonopoulos , V. a. Dougalis , G. Kounadis

We propose and analyse a fully-discrete discontinuous Galerkin time-stepping method for parabolic Hamilton--Jacobi--Bellman equations with Cordes coefficients. The method is consistent and unconditionally stable on rather general…

Numerical Analysis · Mathematics 2017-03-16 Iain Smears , Endre Süli

This paper generalizes the earlier work on the energy-based discontinuous Galerkin method for second-order wave equations to fourth-order semilinear wave equations. We first rewrite the problem into a system with a second-order spatial…

Numerical Analysis · Mathematics 2022-07-25 Lu Zhang

Stochastic wave equations appear in several models for evolutionary processes subject to random forces, such as the motion of a strand of DNA in a liquid or heat flow around a ring. Semilinear stochastic wave equations can typically not be…

Probability · Mathematics 2021-11-09 Ladislas Jacobe de Naurois , Arnulf Jentzen , Timo Welti

We consider the dynamic Biot model (see [Biot, M. A. J. Appl. Phys. 33, 1482--1498 (1962)]) describing the interaction between fluid flow and solid deformation including wave propagation phenomena in both the liquid and solid phases of a…

Numerical Analysis · Mathematics 2025-07-29 Johannes Kraus , Maria Lymbery , Kevin Osthues

We present a numerical comparison between two standard finite volume schemes and a discontinuous Galerkin method applied to the BGK equation of rarefied gas dynamics. We pay a particular attention to the numerical boundary conditions in…

Numerical Analysis · Mathematics 2020-03-13 Céline Baranger , Nicolas Hérouard , Julien Mathiaud , Luc Mieussens

We address the Riemann and Cauchy problems for systems of $n$ conservation laws in $m$ unknowns which are subject to $m-n$ constraints ($m\geq n$). Such constrained systems generalize systems of conservation laws in standard form to include…

Mathematical Physics · Physics 2017-09-28 Moritz Reintjes

In this paper we establish best approximation property of fully discrete Galerkin solutions of second order parabolic problems on convex polygonal and polyhedral domains in the $L^\infty(I;W^{1,\infty}(\Om))$ norm. The discretization method…

Numerical Analysis · Mathematics 2018-08-20 Dmitriy Leykekhman , Boris Vexler

The paper is concerned with the numerical approximation of the Intermediate Long Wave and Benjamin-Ono systems, that serve as models for the propagation of interfacial internal waves in a two-layer fluid system in particular physical…

Numerical Analysis · Mathematics 2021-04-21 Vassilios A. Dougalis , Angel Durán , Leetha Saridaki

Finite element methods provide accurate and efficient methods for the numerical solution of partial differential equations by means of restricting variational problems to finite-dimensional approximating spaces. However, they do not…

Numerical Analysis · Mathematics 2025-06-24 Robert C. Kirby , John D. Stephens

The problem of late time instability in time domain integral equations for electromagnetics is longstanding. While several techniques have been suggested for addressing this problem, they either require impractically high degrees of freedom…

Mathematical Physics · Physics 2012-12-13 N. V. Nair , A. J. Pray , B. Shanker

In this article, we present an Unfitted Space-Time Finite Element method for the scalar transport equation posed on moving domains. We consider the case of the domain boundary being transported by the same velocity field as the scalar…

Numerical Analysis · Mathematics 2025-09-03 Erik Burman , Fabian Heimann

A new modified Galerkin / Finite Element Method is proposed for the numerical solution of the fully nonlinear shallow water wave equations. The new numerical method allows the use of low-order Lagrange finite element spaces, despite the…

Numerical Analysis · Mathematics 2016-09-21 Dimitrios Mitsotakis , Costas Synolakis , Mark Mcguinness

We introduce a hybrid method to couple continuous Galerkin finite element methods and high-order finite difference methods in a nonconforming multiblock fashion. The aim is to optimize computational efficiency when complex geometries are…

Numerical Analysis · Mathematics 2021-11-24 Tuan Anh Dao , Ken Mattsson , Murtazo Nazarov