Related papers: Bounds-constrained finite element approximation of…
We show strong uniform convergence of monotone P1 finite element methods to the viscosity solution of isotropic parabolic Hamilton-Jacobi-Bellman equations with mixed boundary conditions on unstructured meshes and for possibly degenerate…
One of main obstacles in verifying the energy dissipation laws of implicit-explicit Runge-Kutta (IERK) methods for phase field equations is to establish the uniform boundedness of stage solutions without the global Lipschitz continuity…
This paper investigates, a new class of fractional order Runge-Kutta (FORK) methods for numerical approximation to the solution of fractional differential equations (FDEs). By using the Caputo generalizedTaylor formula and the total…
A space-time fully adaptive multiresolution method for evolutionary non-linear partial differential equations is presented introducing an improved local time-stepping method. The space discretisation is based on classical finite volumes,…
Variational space-time formulations for Partial Differential Equations have been of great interest in the last decades. While it is known that implicit time marching schemes have variational structure, the Galerkin formulation of explicit…
We consider a mixed finite element method for a biharmonic equation with clamped boundary conditions based on biorthogonal systems with weakly imposed Dirichlet boundary condition. We show that the weak imposition of the boundary condition…
The exponential fitting technique uses information on the expected behaviour of the solution of a differential problem to define accurate and efficient numerical methods. In particular, exponentially fitted methods are very effective when…
We construct a finite element method for the numerical solution of a fractional porous medium equation on a bounded open Lipschitz polytopal domain $\Omega \subset \mathbb{R}^{d}$, where $d = 2$ or $3$. The pressure in the model is defined…
We introduce a novel method for bounding high-order multi-dimensional polynomials in finite element approximations. The method involves precomputing optimal piecewise-linear bounding boxes for polynomial basis functions, which can then be…
This paper proposes a finite element method for solving the periodic steady-state problem for the scalar-valued and vector-valued Poisson equations, a simple reduction model of the Maxwell equations under the Coulomb gauge. Introducing a…
In this paper, a symmetrized two-scale finite element method is proposed for a class of partial differential equations with symmetric solutions. With this method, the finite element approximation on a fine tensor product grid is reduced to…
The purpose of this work is to test the application of the finite element method to quantum mechanical problems, in particular for solving the Schroedinger equation. We begin with an overview of quantum mechanics, and standard numerical…
This paper describes in detail the implementation of a finite element technique for solving the compressible Navier-Stokes equations that is provably robust and demonstrates excellent performance on modern computer hardware. The method is…
We consider the efficient numerical solution of coupled dynamical systems, consisting of a small nonlinear part and a large linear time invariant part, possibly stemming from spatial discretization of an underlying partial differential…
New variational formulations are devised for the curl--div system, and the corresponding finite element approximations are shown to converge. Curl--free and divergence--free finite elements are employed for discretizing the problem.
We propose and study a scheme combining the finite element method and machine learning techniques for the numerical approximations of coupled nonlinear forward-backward stochastic partial differential equations (FBSPDEs) with homogeneous…
We derive mixed finite element discretizations of a cold relativistics fluid model from approximations of the Poisson bracket that preserve mass, energy and the divergence constraints. For time-discretization we derive an implicit…
To achieve efficient and accurate long-time integration, we propose a fast, accurate, and stable high-order numerical method for solving fractional-in-space reaction-diffusion equations. The proposed method is explicit in nature and…
We identify the stochastic processes associated with one-sided fractional partial differential equations on a bounded domain with various boundary conditions. This is essential for modelling using spatial fractional derivatives. We show…
The EHP and the MCAP provide new rigorous weak variational formalism for a broad range of initial boundary value problems in mathematical physics and mechanics. Both approaches utilize the mixed formulation and lead to the development of…