We derive a residual based a-posteriori error estimate for the outer normal flux of approximations to {the diffusion problem with variable coefficient}. By analyzing the solution of the adjoint problem, we show that error indicators in the bulk may be defined to be of higher order than those close to the boundary, which lead to more economic meshes. The theory is illustrated with some numerical examples.
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.
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…
In this paper we consider a sub-diffusion problem where the fractional time derivative is approximated either by the L1 scheme or by Convolution Quadrature. We propose new interpretations of the numerical schemes which lead to a posteriori error estimates. Our approach is based on appropriate pointwise representations of the numerical schemes as perturbed evolution equations and on stability estimates for the evolution equation. A posteriori error estimates in $L^2(H)$ and $L^\infty (H)$ norms of optimal order are derived. Extensive numerical experiments indicate the reliability and the optima…
This paper considers the finite element solution of the boundary value problem of Poisson's equation and proposes a guaranteed em a posteriori local error estimation based on the hypercircle method. Compared to the existing literature on qualitative error estimation, the proposed error estimation provides an explicit and sharp bound for the approximation error in the subdomain of interest, and its efficiency can be enhanced by further utilizing a non-uniform mesh. Such a result is applicable to problems without $H^2$-regularity, since it only utilizes the first order derivative of the solution…
An initial-boundary value problem of subdiffusion type is considered; the temporal component of the differential operator has the form $\sum_{i=1}^{\ell}q_i(t)\, D _t ^{α_i} u(x,t)$, where the $q_i$ are continuous functions, each $D _t ^{α_i}$ is a Caputo derivative, and the $α_i$ lie in $(0,1]$. Maximum/comparison principles for this problem are proved under weak hypotheses. A new positivity result for the multinomial Mittag-Leffler function is derived. A posteriori error bounds are obtained in $L_2(Ω)$ and $L_\infty(Ω)$, where the spatial domain $Ω$ lies in $\bR^d$ with $d\in\{1,2,3\}$. An a…
A posteriori error analysis and adaptivity of a space-time finite element method for the wave equation in second order formulation
Inria（法国国家信息与自动化研究所）；CERMICS, Ecole nationale des ponts et chaussées, IP Paris, CNRS（国立桥路学校CERMICS实验室，巴黎理工学院，法国国家科学研究中心）；The Maxwell Institute for Mathematical Sciences and Department of Mathematics, Heriot-Watt University（赫瑞-瓦特大学麦克斯韦数学科学研究所与数学系）；Department of Mathematics, School of Applied Mathematical and Physical Sciences, National Technical University of Athens（雅典国立工业大学应用数理科学学院数学系）；IACM-FORTH（希腊计算与应用数学研究所）；Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca（米兰比可卡大学数学与应用科学系）；IMATI-CNR（意大利国家研究委员会应用数学研究所）；Fakultät für Mathematik, Universität Wien（维也纳大学数学系）
We prove residual-type a posteriori error estimates in the maximum norm for a linear scalar elliptic convection-diffusion problem that may be singularly perturbed. Similar error analysis in the energy norm by Verfürth indicates that a dual norm of the {convective derivative of the} error must be added to the natural energy norm in order for the natural residual estimator to be reliable and efficient. We show that the situation is similar for the maximum norm. In particular, we define a mesh-dependent weighted seminorm of the convective error which functions as a maximum-norm counterpart to the…
In this paper, we develop an adaptive finite element method for the nonlinear steady-state Poisson-Nernst-Planck equations, where the spatial adaptivity for geometrical singularities and boundary layer effects are mainly considered. As a key contribution, the steady-state Poisson-Nernst-Planck equations are studied systematically and rigorous analysis for a residual-based a posteriori error estimate of the nonlinear system is presented. With the help of Schauder fixed point theorem, we show the solution existence and uniqueness of the linearized system derived by taking $G-$derivatives of the …
Dimension reduction for derivative-informed operator learning: An analysis of approximation errors
Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin（德克萨斯大学奥斯汀分校奥登计算工程与科学研究所）；School of Computational Science and Engineering, Georgia Institute of Technology（佐治亚理工学院计算科学与工程学院）；Walker Department of Mechanical Engineering, The University of Texas at Austin（德克萨斯大学奥斯汀分校沃克机械工程系）
We develop the theory of fractional gradient flows: an evolution aimed at the minimization of a convex, l.s.c.~energy, with memory effects. This memory is characterized by the fact that the negative of the (sub)gradient of the energy equals the so-called Caputo derivative of the state. We introduce the notion of energy solutions, for which we provide existence, uniqueness and certain regularizing effects. We also consider Lipschitz perturbations of this energy. For these problems we provide an a posteriori error estimate and show its reliability. This estimate depends only on the problem data,…
