Related papers: Eulerian and Semi-Lagrangian Methods for Convectio…
Using the asymmetric fractional calculus of variations, we derive a fractional Lagrangian variational formulation of the convection-diffusion equation in the special case of constant coefficients.
We derive optimal $L^2$-error estimates for semilinear time-fractional subdiffusion problems involving Caputo derivatives in time of order $\alpha\in (0,1)$, for cases with smooth and nonsmooth initial data. A general framework is…
A Streamline Upwind Petrov-Galerkin (SUPG) finite element method for transient convection-diffusion-reaction equation in time-dependent domains is proposed. In particular, a convection dominated transient scalar problem is considered. The…
We study a model of crowd motion following a gradient vector field, with possibly additional interaction terms such as attraction/repulsion, and we present a numerical scheme for its solution through a Lagrangian discretization. The density…
We propose a new Eulerian-Lagrangian (EL) discontinuous Galerkin (DG) method. The method is designed as a generalization of the semi-Lagrangian (SL) DG method for linear advection problems proposed in [J. Sci. Comput. 73: 514-542, 2017],…
In this paper, we present a numerical technique for performing Lie advection of arbitrary differential forms. Leveraging advances in high-resolution finite volume methods for scalar hyperbolic conservation laws, we first discretize the…
A Lagrangian numerical scheme for solving nonlinear degenerate Fokker-Planck equations in space dimensions $d\ge2$ is presented. It applies to a large class of nonlinear diffusion equations, whose dynamics are driven by internal energies…
In this paper we discuss the local discontinuous Galerkin methods coupled with two specific explicit-implicit-null time discretizations for solving one-dimensional nonlinear diffusion problems $U_t=(a(U)U_x)_x$. The basic idea is to add and…
The paper introduces a finite element method for an Eulerian formulation of partial differential equations governing the transport and diffusion of a scalar quantity in a time-dependent domain. The method follows the idea from Lehrenfeld &…
It is well known that certain fractional diffusion equations can be solved by the densities of stable L\'evy motions. In this paper we use the classical semigroup approach for L\'evy processes to define semi-fractional derivatives, which…
A discrete version of Lagrangian reduction is developed in the context of discrete time Lagrangian systems on $G\times G$, where $G$ is a Lie group. We consider the case when the Lagrange function is invariant with respect to the action of…
We present a fully discrete scheme for the numerical approximation of a moving-boundary problem describing diffusants penetration into rubber. Our scheme utilizes the Galerkin finite element method for the space discretization combined with…
In this paper, we consider control constrained $L^2-$Dirichlet boundary control of a convection-diffusion equation on a two dimensional convex polygonal domain. We discretize the control problem based on the local discontinuous Galerkin…
This work presents the discontinuous Galerkin discretization of the consistent splitting scheme proposed by Liu [J. Liu, J. Comp. Phys., 228(19), 2009]. The method enforces the divergence-free constraint implicitly, removing…
We propose a new family of high order staggered semi-implicit discontinuous Galerkin (DG) methods for the simulation of natural convection problems. Assuming small temperature fluctuations, the Boussinesq approximation is valid and the flow…
In this work, we develop variational formulations of Petrov-Galerkin type for one-dimensional fractional boundary value problems involving either a Riemann-Liouville or Caputo derivative of order $\alpha\in(3/2, 2)$ in the leading term and…
We present a new discretization for advection-diffusion problems with Robin boundary conditions on complex time-dependent domains. The method is based on second order cut cell finite volume methods introduced by Bochkov et al. to discretize…
In convection-dominated flows, robustness of the spatial discretisation is a key property. While Interior Penalty Galerkin (IPG) methods already proved efficient in the situation of large mesh Peclet numbers, Arbitrary Lagrangian-Eulerian…
We introduce a new framework of numerical multiscale methods for advection-dominated problems motivated by climate sciences. Current numerical multiscale methods (MsFEM) work well on stationary elliptic problems but have difficulties when…
We analyze a space-time hybridizable discontinuous Galerkin method to solve the time-dependent advection-diffusion equation on deforming domains. We prove stability of the discretization in the advection-dominated regime by using weighted…