We prove first-order convergence of the semi-explicit Euler scheme combined with a finite element discretization in space for elliptic-parabolic problems which are weakly coupled. This setting includes poroelasticity, thermoelasticity, as well as multiple-network models used in medical applications. The semi-explicit approach decouples the system such that each time step requires the solution of two small and well-structured linear systems rather than the solution of one large system. The decoupling improves the computational efficiency without decreasing the convergence rates. The presented convergence proof is based on an interpretation of the scheme as an implicit method applied to a constrained partial differential equation with delay term. Here, the delay time equals the used step size. This connection also allows a deeper understanding of the weak coupling condition, which we accomplish to quantify explicitly.
This paper studies the convergence of a spatial semi-discretization for a backward semilinear stochastic parabolic equation. The filtration is general, and the spatial semi-discretization uses the standard continuous piecewise linear element method. Firstly, higher regularity of the solution to the continuous equation is derived. Secondly, the first-order spatial accuracy is derived for the spatial semi-discretization. Thirdly, an application of the theoretical result to a stochastic linear quadratic control problem is presented.
This article deals with a simultaneous abstract evolution equation. This includes a parabolic-hyperbolic phase-field system as an example which consists of a parabolic equation for the relative temperature coupled with a semilinear damped wave equation for the order parameter. Although a time discretization of an initial value problem for an abstract evolution equation has been studied, time discretizations of initial value problems for simultaneous abstract evolution equations seem to be not studied yet. In this paper we focus on a time discretization of a simultaneous abstract evolution equa…
We analyze Galerkin discretizations of a new well-posed mixed space-time variational formulation of parabolic PDEs. For suitable pairs of finite element trial spaces, the resulting Galerkin operators are shown to be uniformly stable. The method is compared to two related space-time discretization methods introduced in [IMA J. Numer. Anal., 33(1) (2013), pp. 242-260] by R. Andreev and in [Comput. Methods Appl. Math., 15(4) (2015), pp. 551-566] by O. Steinbach.
We consider a minimal residual discretization of a simultaneous space-time variational formulation of parabolic evolution equations. Under the usual `LBB' stability condition on pairs of trial- and test spaces we show quasi-optimality of the numerical approximations without assuming symmetry of the spatial part of the differential operator. Under a stronger LBB condition we show error estimates in an energy-norm which are independent of this spatial differential operator.
Runge-Kutta time discretization of nonlinear parabolic equations studied via discrete maximal parabolic regularity
Institut für Analysis, KIT（卡尔斯鲁厄理工学院分析与研究所）；Department of Applied Mathematics, The Hong Kong Polytechnic University（香港理工大学应用数学系）；Mathematisches Institut, Universität Tübingen（蒂宾根大学数学研究所）
In this paper we consider the numerical approximation of a general second order semi-linear parabolic partial differential equation. Equations of this type arise in many contexts, such as transport in porous media. Using finite element method for space discretization and the exponential Rosenbrock-Euler method for time discretization, we provide a rigorous convergence proof in space and time under only the standard Lipschitz condition of the nonlinear part for both smooth and nonsmooth initial solution. This is in contrast to very restrictive assumptions made in the literature, where the autho…
This paper is concerned with the approximation of linear and nonlinearinitial-boundary-value problems of pseudo-parabolic equations with Dirichlet boundary conditions. They are discretized in space by spectral Galerkin and collocation methods based on Legendre and Chebyshev polynomials. The time integration is carried out suitably with robust schemes attending to qualitative features such as stiffness and preservation of strong stability to simulate nonregular problems more correctly. The corresponding semidiscrete and fully discrete schemes are described and the performance of the methods is …
A discretization of an optimal control problem of a stochastic parabolic equation driven by multiplicative noise is analyzed. The state equation is discretized by the continuous piecewise linear element method in space and by the backward Euler scheme in time. The convergence rate $ O(τ^{1/2} + h^2) $ is rigorously derived.
We investigate the resolution of parabolic PDEs via Extreme Learning Machine (ELMs) Neural Networks, which have a single hidden layer and can be trained at a modest computational cost as compared with Deep Learning Neural Networks. Our approach addresses the time evolution by applying classical ODEs techniques and uses ELM-based collocation for solving the resulting stationary elliptic problems. In this framework, the $θ$-method and Backward Difference Formulae (BDF) techniques are investigated on some linear parabolic PDEs that are challeging problems for the stability and accuracy properties…
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…
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…
This paper investigates the influences of standard numerical discretizations on hitting probabilities for linear stochastic parabolic system driven by space-time white noises. We establish lower and upper bounds for hitting probabilities of the associated numerical solutions of both temporal and spatial semi-discretizations in terms of Bessel-Riesz capacity and Hausdorff measure, respectively. Moreover, the critical dimensions of both temporal and spatial semi-discretizations turn out to be half of those of the exact solution. This reveals that for a large class of Borel sets $A$, the probabil…
Maximal parabolic $L^p$-regularity of linear parabolic equations on an evolving surface is shown by pulling back the problem to the initial surface and studying the maximal $L^p$-regularity on a fixed surface. By freezing the coefficients in the parabolic equations at a fixed time and utilizing a perturbation argument around the freezed time, it is shown that backward difference time discretizations of linear parabolic equations on an evolving surface along characteristic trajectories can preserve maximal $L^p$-regularity in the discrete setting. The result is applied to prove the stability an…
For linear parabolic initial-boundary value problems with self-adjoint, time-homogeneous elliptic spatial operator in divergence form with Lipschitz-continuous coefficients, and for incompatible, time-analytic forcing term in polygonal/polyhedral domains $D$, we prove time-analyticity of solutions. Temporal analyticity is quantified in terms of weighted, analytic function classes, for data with finite, low spatial regularity and without boundary compatibility. Leveraging this result, we prove exponential convergence of a conforming, semi-discrete $hp$-time-stepping approach. We combine this se…
Time discretizations of phase-field systems have been studied. For example, a time discretization and an error estimate for a parabolic-parabolic phase-field system have been studied by Colli--K. [Commun. Pure Appl. Anal. 18 (2019)]. Also, a time discretization and an error estimate for a simultaneous abstract evolution equation applying parabolic-hyperbolic phase field systems and the linearized equations of coupled sound and heat flow have been studied (see K. [ESAIM Math. Model. Numer. Anal.54 (2020), Electron. J. Differential Equations 2020, Paper No. 96]). On the other hand, although exis…
We present an algorithm for the solution of a simultaneous space-time discretization of linear parabolic evolution equations with a symmetric differential operator in space. Building on earlier work, we recast this discretization into a Schur-complement equation whose solution is a quasi-optimal approximation to the weak solution of the equation at hand. Choosing a tensor-product discretization, we arrive at a remarkably simple linear system. Using wavelets in time and standard finite elements in space, we solve the resulting system in linear complexity on a single processor, and in polylogari…
In this paper we deal with an abstract problem which includes the linearized equations of coupled sound and heat flow as an example. Recently, a time discretization of a simultaneous abstract evolution equation applying to some parabolic-hyperbolic phase-field systems has been studied. This paper focuses on a time discretization of an abstract problem applying to the linearized equations of coupled sound and heat flow. Also, this paper gives some parabolic-hyperbolic phase-field systems as examples.
In this work, we present a novel error analysis for recovering a spatially dependent diffusion coefficient in an elliptic or parabolic problem. It is based on the standard regularized output least-squares formulation with an $H^1(Ω)$ seminorm penalty, and then discretized using the Galerkin finite element method with conforming piecewise linear finite elements for both state and coefficient, and backward Euler in time in the parabolic case. We derive \textit{a priori} weighted $L^2(Ω)$ estimates where the constants depend only on the given problem data for both elliptic and parabolic cases. Fu…
Boundary value problem for a fractional power of an elliptic operator is considered. An integral representation by means of a standard solution problem for parabolic equations is used to solve such problems. Quadrature generalized Gauss-Laguerre formulas are constructed. We examine the effect of key parameters on the accuracy of the approximate solution: the number of nodes of the quadrature and fractional power of the operator. Computational experiments were performed to model two-dimensional problem with a fractional power of an elliptic operator.
In order to characterize the fluctuation between the ergodic limit and the time-averaging estimator of a full discretization in a quantitative way, we establish a central limit theorem for the full discretization of the parabolic stochastic partial differential equation. The theorem shows that the normalized time-averaging estimator converges to a normal distribution with the variance being the same as that of the continuous case, where the scale used for the normalization corresponds to the temporal strong convergence order of the considered full discretization. A key ingredient in the proof …
We study the discretization of a linear evolution partial differential equation when its Green function is known. We provide error estimates both for the spatial approximation and for the time stepping approximation. We show that, in fact, an approximation of the Green function is almost as good as the Green function itself. For suitable time-dependent parabolic equations, we explain how to obtain good, explicit approximations of the Green function using the Dyson-Taylor commutator method (DTCM) that we developed in J. Math. Phys. (2010). This approximation for short time, when combined with a…
This paper deals with the backward Euler method applied to semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive noise. The SPDE is discretized in space by the finite element method and in time by the backward Euler. We consider a larger class of nonlinear drift functions, which are of Nemytskii type and polynomial of any odd degree with negative leading term, instead of only dealing with the special case of stochastic Allen-Chan equation as in the up to date literature.Moreover our linear operator is of second order and not necessarily self-adjoint, therefo…
A Stable SPH Discretization of the Elliptic Operator with Heterogeneous Coefficients
Delft University of Technology, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft Institute of Applied Mathematics（代尔夫特理工大学，电气、数学与计算机科学学院，应用数学研究所）
We analyze space-time finite element methods for the numerical solution of distributed parabolic optimal control problems with energy regularization in the Bochner space $L^2(0,T;H^{-1}(Ω))$. By duality, the related norm can be evaluated by means of the solution of an elliptic quasi-stationary boundary value problem. When eliminating the control, we end up with the reduced optimality system that is nothing but the variational formulation of the coupled forward-backward primal and adjoint equations. Using Babuška's theorem, we prove unique solvability in the continuous case. Furthermore, we est…
A semi-Lagrangian method for parabolic problems is proposed, that extends previous work by the authors to achieve a fully conservative, flux-form discretization of linear and nonlinear diffusion equations. A basic consistency and convergence analysis are proposed. Numerical examples validate the proposed method and display its potential for consistent semi-Lagrangian discretization of advection--diffusion and nonlinear parabolic problems.
