Related papers: A comparison study of deep Galerkin method and dee…
In recent years a large literature on deep learning based methods for the numerical solution partial differential equations has emerged; results for integro-differential equations on the other hand are scarce. In this paper we study deep…
The Galerkin method is often employed for numerical integration of evolutionary equations, such as the Navier-Stokes equation or the magnetic induction equation. Application of the method requires solving an equation of the form $P(Av-f)=0$…
In this paper, we revisit the nonoverlapping domain decomposition methods for solving elliptic problems with high contrast coefficients. Some interesting results are discovered. We find that the Dirichlet-Neumann algorithm and Robin-Robin…
Deep learning-based partial differential equation(PDE) solvers have received much attention in the past few years. Methods of this category can solve a wide range of PDEs with high accuracy, typically by transforming the problems into…
In recent years, neural networks have achieved remarkable progress in various fields and have also drawn much attention in applying them on scientific problems. A line of methods involving neural networks for solving partial differential…
This work introduces finite element methods for a class of elliptic fully nonlinear partial differential equations. They are based on a minimal residual principle that builds upon the Alexandrov--Bakelman--Pucci estimate. Under rather…
In this article we consider two-grid finite element methods for solving semilinear interface problems in d space dimensions, for d=2 or d=3. We first describe in some detail the target problem class with discontinuous diffusion…
In this paper, we consider a class of discontinuous Galerkin (DG) methods for one-dimensional nonlocal diffusion (ND) problems. The nonlocal models, which are integral equations, are widely used in describing many physical phenomena with…
Partial differential equations have a wide range of applications in modeling multiple physical, biological, or social phenomena. Therefore, we need to approximate the solutions of these equations in computationally feasible terms. Nowadays,…
Numerical climate- and weather-prediction requires the fast solution of the equations of fluid dynamics. Discontinuous Galerkin (DG) discretisations have several advantageous properties. They can be used for arbitrary domains and support a…
In this study, we examine numerical approximations for 2nd-order linear-nonlinear differential equations with diverse boundary conditions, followed by the residual corrections of the first approximations. We first obtain numerical results…
In this paper, we study adaptive neuron enhancement (ANE) method for solving self-adjoint second-order elliptic partial differential equations (PDEs). The ANE method is a self-adaptive method generating a two-layer spline NN and a numerical…
In this paper we establish best approximation property of fully discrete Galerkin solutions of second order parabolic problems on convex polygonal and polyhedral domains in the $L^\infty(I;W^{1,\infty}(\Om))$ norm. The discretization method…
In this paper we investigate the application of pseudo-transient-continuation (PTC) schemes for the numerical solution of semilinear elliptic partial differential equations, with possible singular perturbations. We will outline a residual…
Elliptic equations play a crucial role in turbulence models for magnetic confinement fusion. Regardless of the chosen modeling approach - whether gyrokinetic, gyrofluid, or drift-fluid - the Poisson equation and Amp\`{e}re's law lead to…
We present and analyze a discontinuous Galerkin method for the numerical solution of a class of second-order linear mixed-type partial differential equations, i.e. equations that change their nature from elliptic to hyperbolic through the…
The embedded discontinuous Galerkin (EDG) method by Cockburn et al. [SIAM J. Numer. Anal., 2009, 47(4), 2686-2707] is obtained from the hybridizable discontinuous Galerkin method by changing the space of the Lagrangian multiplier from…
We present a constructive method to devise boundary conditions for solutions of second-order elliptic equations so that these solutions satisfy specific qualitative properties such as: (i) the norm of the gradient of one solution is bounded…
Option pricing often requires solving partial differential equations (PDEs). Although deep learning-based PDE solvers have recently emerged as quick solutions to this problem, their empirical and quantitative accuracy remain not well…
In this article we investigate a finite element formulation of strongly monotone quasi-linear elliptic PDEs in the context of fixed-point iterations. As opposed to Newton's method, which requires information from the previous iteration in…