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We present a novel approach for solving the shallow water equations using a discontinuous Galerkin spectral element method. The method we propose has three main features. First, it enjoys a discrete well-balanced property, in a spirit…
Hyperbolic-parabolic partial differential equations are widely used for the modeling of complex, multiscale problems. High-order methods such as the discontinuous Galerkin (DG) scheme are attractive candidates for their numerical…
Eddy-current problems occur in a wide range of industrial and metallurgical applications where conducting material is processed inductively. Motivated by realising coupled multi-physics simulations, we present a new method for the solution…
In this paper we perform, by means of Discontinuous Galerkin (DG) Finite Element Method (FEM) based numerical solvers for Boltzmann-Poisson (BP) semiclassical models of hot electronic transport in semiconductors, a numerical study of…
We consider a semilinear parabolic equation with a large class of nonlinearities without any growth conditions. We discretize the problem with a discontinuous Galerkin scheme dG(0) in time (which is a variant of the implicit Euler scheme)…
Numerically solving high-dimensional random parametric PDEs poses a challenging computational problem. It is well-known that numerical methods can greatly benefit from adaptive refinement algorithms, in particular when functional…
In this paper we present an advanced numerical method to simulate a real life challenging industrial problem that consists of the non-destructive testing in steam generators. We develop a finite element technique that handles the big data…
In this article, the piecewise-linear finite element method (FEM) is applied to approximate the solution of time-fractional diffusion equations on bounded convex domains. Standard energy arguments do not provide satisfactory results for…
The aim of this paper is to apply a high-order discontinuous-in-time scheme to second-order hyperbolic partial differential equations (PDEs). We first discretize the PDEs in time while keeping the spatial differential operators…
In this article, an efficient transient electricalthermal co-simulation method based on the finite element method (FEM) and the discontinuous Galerkin time-domain (DGTD) method is developed for electrical-thermal coupling analysis of…
The finite element method (FEM) is applied to obtain numerical solutions to a recently derived nonlinear equation for the shallow water wave problem. A weak formulation and the Petrov-Galerkin method are used. It is shown that the FEM gives…
We develop a family of cut finite element methods of different orders based on the discontinuous Galerkin framework, for hyperbolic conservation laws with stationary interfaces in both one and two space dimensions, and for moving interfaces…
In this paper, we propose and analyze a numerically stable and convergent scheme for a convection-diffusion-reaction equation in the convection-dominated regime. Discontinuous Galerkin (DG) methods are considered since standard finite…
We present and analyze a discontinuous Galerkin method for the numerical modelling of the non-linear fully-coupled thermo-poroelastic problem. For the spatial discretization, we design a high-order discontinuous Galerkin method on polygonal…
This is a survey on the theory of adaptive finite element methods (AFEMs), which are fundamental in modern computational science and engineering but whose mathematical assessment is a formidable challenge. We present a self-contained and…
We consider the initial boundary value problem for the homogeneous time-fractional diffusion equation $\partial^\alpha_t u - \De u =0$ ($0< \alpha < 1$) with initial condition $u(x,0)=v(x)$ and a homogeneous Dirichlet boundary condition in…
A novel boundary element formulation for solving problems involving eddy currents in the thin skin depth approximation is developed. It is assumed that the time-harmonic magnetic field outside the scatterers can be described using the…
An integro-differential equation, modeling dynamic fractional order viscoelasticity, with a Mittag-Leffler type convolution kernel is considered. A discontinuous Galerkin method, based on piecewise constant polynomials is formulated for…
There has been an arising trend of adopting deep learning methods to study partial differential equations (PDEs). This article is to propose a Deep Learning Galerkin Method (DGM) for the closed-loop geothermal system, which is a new coupled…
This paper presents a space-time interface-fitted finite element method for solving a parabolic advection-diffusion problem with a nonstationary interface. The jumping diffusion coefficient gives rise to the discontinuity of the solution…