Related papers: A posteriori error analysis for a space-time paral…
The paper is concerned with parabolic time-periodic boundary value problems which are of theoretical interest and arise in different practical applications. The multiharmonic finite element method is well adapted to this class of parabolic…
Multilevel methods represent a powerful approach in numerical solution of partial differential equations. The multilevel structure can also be used to construct estimates for total and algebraic errors of computed approximations. This paper…
In two and three dimensional Lipschitz, but not necessarily convex, polytopal domains, we propose and analyze a posteriori error estimators for an optimal control problem involving the stationary Navier--Stokes equations; control…
We investigate the application of a posteriori error estimates to a fractional optimal control problem with pointwise control constraints. Specifically, we address a problem in which the state equation is formulated as an integral form of…
We present original time-parallel algorithms for the solution of the implicit Euler discretization of general linear parabolic evolution equations with time-dependent self-adjoint spatial operators. Motivated by the inf-sup theory of…
We propose a simple domain decomposition method for $d$-dimensional elliptic PDEs which involves an overlapping decomposition into local subdomain problems and a global coarse problem. It relies on a space-filling curve to create equally…
In a general setting, we study a posteriori estimates used in finite element analysis to measure the error between a solution and its approximation. The latter is not necessarily generated by a finite element method. We show that the error…
To enhance solution accuracy and training efficiency in neural network approximation to partial differential equations, partitioned neural networks can be used as a solution surrogate instead of a single large and deep neural network…
We propose a primal-dual parallel proximal splitting method for solving domain decomposition problems for partial differential equations. The problem is formulated via minimization of energy functions on the subdomains with coupling…
The purpose of this article is to propose ODE based approaches for the numerical evaluation of matrix functions $f(A)$, a question of major interest in the numerical linear algebra. To this end, we model $f(A)$ as the solution at a finite…
In this work, we propose an easy-to-implement fixed-point algorithm for reconstructing a space-time dependent source in a subdiffusion model from lateral boundary measurements. The numerical scheme combines a Galerkin finite element method…
We propose a time-space discretization scheme for quasi-linear parabolic PDEs. The algorithm relies on the theory of fully coupled forward--backward SDEs, which provides an efficient probabilistic representation of this type of equation.…
Strong approximation errors of both finite element semi-discretization and spatio-temporal full discretization are analyzed for the stochastic Allen-Cahn equation driven by additive noise in space dimension $d \leq 3$. The full…
This work continues a line of works on developing partially explicit methods for multiscale problems. In our previous works, we have considered linear multiscale problems, where the spatial heterogeneities are at subgrid level and are not…
The ParaOpt algorithm was recently introduced as a time-parallel solver for optimal-control problems with a terminal-cost objective, and convergence results have been presented for the linear diffusive case with implicit-Euler time…
Many multiscale problems have a high contrast, which is expressed as a very large ratio between the media properties. The contrast is known to introduce many challenges in the design of multiscale methods and domain decomposition…
A concise Matlab implementation of a stable parallelizable space-time Petrov-Galerkin discretization for parabolic evolution equations is given. Emphasis is on reusability of spatial finite element codes.
For the pure biharmonic equation and a biharmonic singular perturbation problem, a residual-based error estimator is introduced which applies to many existing nonconforming finite elements. The error estimator involves the local…
This paper develops a framework for the error analysis in nonparametric model fitting of fractional stochastic differential equations based on discrete observations. We identify and quantify the main error sources -- time discretization,…
In this paper, we propose a certified reduced basis (RB) method for quasilinear parabolic problems. The method is based on a space-time variational formulation. We provide a residual-based a-posteriori error bound on a space-time level and…