相关论文: Further results on a space-time FOSLS formulation …

In this work, we show that the space-time first-order system least-squares (FOSLS) formulation [F\"uhrer, Karkulik, Comput. Math. Appl. 92 (2021)] for the heat equation and its recent generalization [Gantner, Stevenson, ESAIM Math. Model.…

We consider the first-order system space-time formulation of the heat equation introduced in [Bochev, Gunzburger, Springer, New York (2009)], and analyzed in [F\"uhrer, Karkulik, Comput. Math. Appl. 92 (2021)] and [Gantner, Stevenson, ESAIM…

We present a space-time least squares finite element method for the heat equation. It is based on residual minimization in L2 norms in space-time of an equivalent first order system. This implies that (i) the resulting bilinear form is…

We develop Aubin--Nitsche-type estimates for recently proposed first-order system least-squares finite element methods (FOSLS) for the heat equation. Under certain assumptions, which are satisfied if the spatial domain is convex and the…

We develop couplings of a recent space-time first-order system least-squares (FOSLS) method for parabolic problems and space-time boundary element methods (BEM) for the heat equation to numerically solve a parabolic transmission problem on…

Backward parabolic equations, such as the backward heat equation, are classical examples of ill-posed problems where solutions may not exist or depend continuously on the data. In this work, we study a least squares finite element method to…

We propose a new first-order-system least squares (FOSLS) finite-element discretization for singularly perturbed reaction-diffusion equations. Solutions to such problems feature layer phenomena, and are ubiquitous in many areas of applied…

Different variants of the method of weighted residual finite element method are used to get a solution for the parabolic heat equation, which is considered to be the model equation for the steady state Navier-Stokes equations. Results show…

In this work, we present a POD-greedy reduced basis method for parabolic partial differential equations (PDEs), based on the least squares space-time formulation proposed in [Hinze, Kahle, Stahl, A least-squares space-time approach for…

This paper analyzes the discrete energy laws associated with first-order system least-squares (FOSLS) discretizations of time-dependent partial differential equations. Using the heat equation and the time-dependent Stokes' equation as…

We propose and analyze a general framework for space-time finite element methods that is based on least-squares finite element methods for solving a first-order reformulation of the thick parabolic obstacle problem. Discretizations based on…

In this paper, we present proofs of the coerciveness of first-order system least-squares methods for general (possibly indefinite) second-order linear elliptic PDEs under a minimal uniqueness assumption. For general linear second-order…

This paper studies adaptive first-order least-squares finite element methods for second-order elliptic partial differential equations in non-divergence form. Unlike the classical finite element method which uses weak formulations of PDEs…

In this paper we present an error analysis of an Eulerian finite element method for solving parabolic partial differential equations posed on evolving hypersurfaces in $\mathbb{R}^d$, $d=2,3$. The method employs discontinuous piecewise…

We introduce a conceptual framework for numerically solving linear elliptic, parabolic, and hyperbolic PDEs on bounded, polytopal domains in euclidean spaces by deep neural networks. The PDEs are recast as minimization of a least-squares…

We provide new insights into the a priori theory for a time-stepping scheme based on least-squares finite element methods for parabolic first-order systems. The elliptic part of the problem is of general reaction-convection-diffusion type.…

This paper develops a first-order system least-squares (FOSLS) formulation for equations of two-phase flow. The main goal is to show that this discretization, along with numerical techniques such as nested iteration, algebraic multigrid,…

We present a first order system least squares (FOSLS) method for the Helmholtz equation at high wave number k, which always deduces Hermitian positive definite algebraic system. By utilizing a non-trivial solution decomposition to the dual…

We propose a First-Order System Least Squares (FOSLS) method based on deep-learning for numerically solving second-order elliptic PDEs. The method we propose is capable of dealing with either variational and non-variational problems, and…

We study spatially semidiscrete and fully discrete finite volume element methods for the homogeneous heat equation with homogeneous Dirichlet boundary conditions and derive error estimates for smooth and nonsmooth initial data. We show that…