Related papers: Well-posed variational formulations of Friedrichs-…

This article presents a new primal-dual weak Galerkin finite element method for the div-curl system with tangential boundary conditions and low-regularity assumptions on the solution. The numerical scheme is based on a weak variational form…

In this paper we discuss the adjoint stabilised finite element method introduced in, E. Burman, Stabilized finite element methods for nonsymmetric, noncoercive and ill-posed problems. Part I: elliptic equations, SIAM Journal on Scientific…

For an elliptic, semilinear differential operator of the form $S(u) = A : D^2 u + b(x, u , Du)$, consider the functional $E_\infty(u) = \mathop{\mathrm{ess \, sup}}_\Omega |S(u)|$. We study minimisers of $E_\infty$ for prescribed boundary…

A recent paper of Arnold, Falk, and Winther [Bull AMS, 47 (2010)] showed that a large class of mixed finite element methods can be formulated naturally on Hilbert complexes, where using a Galerkin-like approach, one solves a variational…

We consider an initial/boundary value problem for one-dimensional fractional-order parabolic equations with a space fractional derivative of Riemann-Liouville type and order $\alpha\in (1,2)$. We study a spatial semidiscrete scheme with the…

This paper discusses the upwinded local discontinuous Galerkin methods for the one-term/multi-term fractional ordinary differential equations (FODEs). The natural upwind choice of the numerical fluxes for the initial value problem for FODEs…

We propose a nodal discontinuous Galerkin method for solving the nonlinear Riesz space fractional Schr\"{o}dinger equation and the strongly coupled nonlinear Riesz space fractional Schr\"{o}dinger equations. These problems have been…

We consider an optimal control problem constrained by a parabolic partial differential equation (PDE) with Robin boundary conditions. We use a well-posed space-time variational formulation in Lebesgue--Bochner spaces with minimal…

We consider perturbed nonlinear ill-posed equations in Hilbert spaces, with operators that are monotone on a given closed convex subset. A simple stable approach is Lavrentiev regularization, but existence of solutions of the regularized…

Discontinuous Petrov Galerkin (DPG) methods are made easily implementable using `broken' test spaces, i.e., spaces of functions with no continuity constraints across mesh element interfaces. Broken spaces derivable from a standard exact…

Employing a phase space which includes the (Riemann-Liouville) fractional derivative of curves evolving on real space, we develop a restricted variational principle for Lagrangian systems yielding the so-called restricted fractional…

While it is known that one can consider the Cauchy problem for evolution equations with Caputo derivatives, the situation for the initial value problems for the Riemann-Liouville derivatives is less understood. In this paper we propose new…

In this paper, we study a class of initial-boundary value problems for the Korteweg-de Vries equation posed on a bounded domain $(0,L)$. We show that the initial-boundary value problem is locally well-posed in the classical Sobolev space…

We introduce a fractional variant of the Cahn-Hilliard equation settled in a bounded domain and with a possibly singular potential. We first focus on the case of homogeneous Dirichlet boundary conditions, and show how to prove the existence…

We propose a new class of finite element approximations to ideal compressible magnetohydrodynamic equations in smooth regime. Following variational approximations developed for fluid models in the last decade, our discretizations are built…

We study the well-posedness of the initial value problem for fully nonlinear evolution equations, $u_{t}=f[u],$ where $f$ may depend on up to the first three spatial derivatives of $u.$ We make three primary assumptions about the form of…

In this paper we study strongly coupled elliptic systems in non-variational form involving fractional Laplace operators. We prove Liouville type theorems and, by mean of the blow-up method, we establish a priori bounds of positive solutions…

A reaction-diffusion equation with power nonlinearity formulated either on the half-line or on the finite interval with nonzero boundary conditions is shown to be locally well-posed in the sense of Hadamard for data in Sobolev spaces. The…

We consider the primal and dual forms of the optimality conditions for PDE-contrained optimization problems arising in Data-Driven Computational Mechanics when specialized to the reaction-diffusion context. Starting with the continuous…

Friedrichs' systems (FS) are symmetric positive linear systems of first-order partial differential equations (PDEs), which provide a unified framework for describing various elliptic, parabolic and hyperbolic semi-linear PDEs such as the…