This paper develops and discusses a residual-based a posteriori error estimator for parabolic surface partial differential equations on closed stationary surfaces. The full discretization uses the surface finite element method in space and the backward Euler method in time. The proposed error indicator bounds the error quantities globally in space from above and below, and globally in time from above and locally from below. Based on the derived error indicator, a space-time adaptive algorithm is proposed. Numerical experiments illustrate and complement the theory.
A class of linear parabolic equations are considered. We give a posteriori error estimates in the maximum norm for a method that comprises extrapolation applied to the backward Euler method in time and finite element discretisations in space. We use the idea of elliptic reconstructions and certain bounds for the Green's function of the parabolic operator.
For time-fractional parabolic equations with a Caputo time derivative of order $α\in(0,1)$, we give pointwise-in-time a posteriori error bounds in the spatial $L_2$ and $L_\infty$ norms. Hence, an adaptive mesh construction algorithm is applied for the L1 method, which yields optimal convergence rates $2-α$ in the presence of solution singularities.
A posteriori error estimates in the maximum norm are studied for various time-semidiscretisations applied to a class of linear parabolic equations. We summarise results from the literature and present some new improved error bounds. Crucial ingredients are certain bounds in the $L_1$ norm for the Green's function associated with the parabolic operator and its derivatives.
Functional a posteriori error estimates for parabolic time-periodic boundary value problems
Johannes Kepler University Linz（林茨约翰·开普勒大学）；V. A. Steklov Institute of Mathematics in St. Petersburg（圣彼得堡V.A.斯捷克洛夫数学研究所）；University of Jyväskylä（于韦斯屈莱大学）；Johann Radon Institute for Computational and Applied Mathematics（约翰·拉顿计算与应用数学研究所）
Time-fractional parabolic equations with a Caputo time derivative are considered. For such equations, we explore and further develop the new methodology of the a-posteriori error estimation and adaptive time stepping proposed in [7]. We improve the earlier time stepping algorithm based on this theory, and specifically address its stable and efficient implementation in the context of high-order methods. The considered methods include an L1-2 method and continuous collocation methods of arbitrary order, for which adaptive temporal meshes are shown to yield optimal convergence rates in the presen…
Parametric elliptic reconstructions and a posteriori error estimates for parabolic partial differential equations with small randomness in a Robin boundary condition
Chalmers University of Technology and University of Gothenburg（查尔姆斯理工大学和哥德堡大学）
Optimal $L^\infty(L^2)$ and $L^1(L^2)$ a posteriori error estimates for the fully discrete approximations of time fractional parabolic differential equations
Functional a posteriori error estimates for time-periodic parabolic optimal control problems
Johannes Kepler University Linz（林茨约翰·开普勒大学）；V. A. Steklov Institute of Mathematics in St. Petersburg（圣彼得堡V.A.斯捷洛夫数学研究所）；University of Jyväskylä（于韦斯屈莱大学）；Johann Radon Institute for Computational and Applied Mathematics（约翰·拉登计算与应用数学研究所）
We consider fully discrete time-space approximations of abstract linear parabolic partial differential equations (PDEs) consisting of an $hp$-version discontinuous Galerkin (DG) time stepping scheme in conjunction with standard (conforming) Galerkin discretizations in space. We derive abstract computable a posteriori error bounds resulting, for instance, in concrete bounds in $L_{\infty}(I;L_2(Ω))$- and $L_{2}(I;H^{1}(Ω))$-type norms when $I$ is the temporal and $Ω$ the spatial domain for the PDE. We base our methodology for the analysis on a novel space-time reconstruction approach. Our appro…
We construct a space-time parallel method for solving parabolic partial differential equations by coupling the Parareal algorithm in time with overlapping domain decomposition in space. The goal is to obtain a discretization consisting of "local" problems that can be solved on parallel computers efficiently. However, this introduces significant sources of error that must be evaluated. Reformulating the original Parareal algorithm as a variational method and implementing a finite element discretization in space enables an adjoint-based a posteriori error analysis to be performed. Through an app…
In two dimensions, we propose and analyze an a posteriori error estimator for finite element approximations of the stationary Navier Stokes equations with singular sources on Lipschitz, but not necessarily convex, polygonal domains. Under a smallness assumption on the continuous and discrete solutions, we prove that the devised error estimator is reliable and locally efficient. We illustrate the theory with numerical examples.
A posteriori error estimates in the maximum norm for parabolic problems
University of Kentucky（肯塔基大学）；University of Sussex（萨塞克斯大学）；University of Crete（克里特大学）；Institute for Applied and Computational Mathematics, Foundation for Research and Technology-Hellas（希腊研究与技术基金会应用与计算数学研究所）
In this paper we develop an a posteriori error analysis for the stationary Stokes-Darcy coupled problem approximated by conforming finite element method on isotropic meshes in $\mathbb{R}^d$, $d\in\{2,3\}$. The approach utilizes a new robust stabilized fully mixed discretization developed by Jiaping Yu et al. (Advances in Difference Equations, SpringerOpen Journal, 2018). The a posteriori error estimate is based on a suitable evaluation on the residual of the finite element solution plus the stabilization terms. It is proven that the a posteriori error estimate provided in this paper is both r…
Guaranteed, locally space-time efficient, and polynomial-degree robust a posteriori error estimates for high-order discretizations of parabolic problems
Magnetohydrodynamics (MHD) is a continuum level model for conducting fluids subject to external magnetic fields, e.g. plasmas and liquid metals. The efficient and robust solution of the MHD system poses many challenges due to it's nonlinear, non self-adjoint, and highly coupled nature. In this paper, we develop a robust and accurate a posteriori error estimate for the numerical solution of the MHD equations based on the exact penalty method. The error estimate also isolates particular contributions of error in a quantity of interest (QoI) to inform discretization choices to arrive at accurate …
We consider the a posteriori error estimation for convection-diffusion-reaction equations in both diffusion-dominated and convection/reaction-dominated regimes. We present an explicit hybrid estimator, which, in each regime, is proved to be reliable and efficient with constants independent of the parameters in the underlying problem. For convection-dominated problems, the norm introduced by Verf{ü}rth \cite{verf2005confusion} is used to measure the approximation error. Various numerical experiments are performed to (1) demonstrate the robustness of the hybrid estimator; (2) show that the hybri…
We consider locally stabilized, conforming finite element schemes on completely unstructured simplicial space-time meshes for the numerical solution of parabolic initial-boundary value problems with variable, possibly discontinuous in space and time, coefficients. Distributional sources are also admitted. Discontinuous coefficients, non-smooth boundaries, changing boundary conditions, non-smooth or incompatible initial conditions, and non-smooth right-hand sides can lead to non-smooth solutions. We present new a priori and a posteriori error estimates for low-regularity solutions. In order to …
Mathematical modeling and numerical multigoal-oriented a posteriori error control and adaptivity for a stationary, nonlinear, coupled flow temperature model with temperature dependent density
A posteriori error estimation and adaptive strategy for PGD model reduction applied to parametrized linear parabolic problems
ENS Cachan/CNRS/Université Paris-Saclay（巴黎高等师范学校卡尚分校/CNRS/巴黎萨克雷大学）；Université Paris-Est, Laboratoire Modélisation et Simulation Multi Echelle, MSME UMR 8208 CNRS（巴黎东大学，多尺度建模与模拟实验室，MSME UMR 8208 CNRS）
Equal order stabilized finite elements with Nitsche for stationary Navier-Stokes problem with slip boundary conditions : a priori and a posteriori error analysis
This paper is concerned with the recovery of (approximate) solutions to parabolic problems from incomplete and possibly inconsistent observational data, given on a time-space cylinder that is a strict subset of the computational domain under consideration. Unlike previous approaches to this and related problems our starting point is a regularized least squares formulation in a continuous infinite-dimensional setting that is based on stable variational time-space formulations of the parabolic PDE. This allows us to derive a priori as well as a posteriori error bounds for the recovered states wi…
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 constraints are also considered. We devise two strategies of discretization: a semi discrete scheme where the control variable is not discretized -- the so-called variational discrezation approach -- and a fully discrete scheme where the control is discretized with piecewise quadratic functions. For each solution solution technique, we design an a posteriori error est…
We present locally stabilized, conforming space-time finite element methods for parabolic evolution equations on hexahedral decompositions of the space-time cylinder. Tensor-product decompositions allow for anisotropic a priori error estimates, that are explicit in spatial and temporal meshsizes. Moreover, tensor-product finite elements are suitable for anisotropic adaptive mesh refinement strategies provided that an appropriate a posteriori discretization error estimator is available. We present such anisotropic adaptive strategies together with numerical experiments.
We present a posteriori error estimates for inconsistent and non-hierarchical Galerkin methods for linear parabolic problems, allowing them to be used in conjunction with very general mesh modification for the first time. We treat schemes which are non-hierarchical in the sense that the spatial Galerkin spaces used on consecutive time-steps may be completely unrelated from one another. The practical interest of this setting is demonstrated by applying our results to finite element methods on moving meshes and using the estimators to drive an adaptive algorithm based on a virtual element method…
The problem of determining the initial condition from noisy final observations in time-fractional parabolic equations is considered. This problem is well-known to be ill-posed and it is regularized by backward Sobolev-type equations. Error estimates of Holder type are obtained with a priori and a posteriori regularization parameter choice rules. The proposed regularization method results in a stable noniterative numerical scheme. The theoretical error estimates are confirmed by numerical tests for one- and two-dimensional equations
We present a robust a posteriori error estimator for the weak Galerkin finite element method applied to stationary convection-diffusion equations in the convection-dominated regime. The estimator provides global upper and lower bounds of the error %measured in a suitable norm and is robust in the sense that upper and lower bounds are uniformly bounded with respect to the diffusion coefficient. Results of the numerical experiments are presented to illustrate the performance of the error estimator.
