Parabolic Integro-differential Equations Research Articles

A Compact FEM Implementation for Parabolic Integro-Differential Equations in 2D

Although two-dimensional (2D) parabolic integro-differential equations (PIDEs) arise in many physical contexts, there is no generally available software that is able to solve them numerically. To remedy this situation, in this article, we provide a compact implementation for solving 2D PIDEs using the finite element method (FEM) on unstructured grids. Piecewise linear finite element spaces on triangles are used for the space discretization, whereas the time discretization is based on the backward-Euler and the Crank–Nicolson methods. The quadrature rules for discretizing the Volterra integral term are chosen so as to be consistent with the time-stepping schemes; a more efficient version of the implementation that uses a vectorization technique in the assembly process is also presented. The compactness of the approach is demonstrated using the software Matrix Laboratory (MATLAB). The efficiency is demonstrated via a numerical example on an L-shaped domain, for which a comparison is possible against the commercially available finite element software COMSOL Multiphysics. Moreover, further consideration indicates that COMSOL Multiphysics cannot be directly applied to 2D PIDEs containing more complex kernels in the Volterra integral term, whereas our method can. Consequently, the subroutines we present constitute a valuable open and validated resource for solving more general 2D PIDEs.

An efficient symmetric finite volume element method for second-order variable coefficient parabolic integro-differential equations

This paper is devoted to develop a symmetric finite volume element (FVE) method to solve second-order variable coefficient parabolic integro-differential equations, arising in modeling of nonlocal reactive flows in porous media. Based on barycenter dual mesh, one semi-discrete and two fully discrete backward Euler and Crank–Nicolson symmetric FVE schemes are presented. Then, the optimal order error estimates in $$L^{2}$$ -norm are derived for the semi-discrete and two fully discrete schemes. Numerical experiments are performed to examine the convergence rate and verify the effectiveness and usefulness of the new numerical schemes.

Global well-posedness for the Euler alignment system with mildly singular interactions

We consider the Euler alignment system with mildly singular interaction kernels. When the local repulsion term is of the fractional type, global in time existence of smooth solutions was proved in [, –]. Here, we consider a class of less singular interaction kernels and establish the global regularity of solutions as long as the interaction kernels are not integrable. The proof relies on modulus of continuity estimates for a class of parabolic integro-differential equations with a drift and mildly singular kernels.

Superclose analysis of a two-grid finite element scheme for semilinear parabolic integro-differential equations

<p style='text-indent:20px;'>In this paper, a two-grid finite element scheme for semilinear parabolic integro-differential equations is proposed. In the two-grid scheme, continuous linear element is used for spatial discretization, while Crank-Nicolson scheme and Leap-Frog scheme are ultilized for temporal discretization. Based on the combination of the interpolation and Ritz projection technique, some superclose estimates between the interpolation and the numerical solution in the <inline-formula><tex-math id="M1">$ H^1 $</tex-math></inline-formula>-norm are derived. Notice that we only need to solve nonlinear problem once in the two-grid scheme, namely, the first time step on the coarse-grid space. A numerical example is presented to verify the effectiveness of the proposed two-grid scheme.</p>

Averaged semi-discrete scheme of sum-approximation for one nonlinear multi-dimensional integro-differential parabolic equation

Abstract The paper is devoted to the construction and study of the additive averaged semi-discrete scheme for one nonlinear multi-dimensional integro-differential equation of parabolic type. The studied equation is based on the well-known Maxwell’s system arising in mathematical simulation of electromagnetic field penetration into a substance.

A Mathematical Model of Tornado

We propose a mathematical model of tornado in the framework of continuum mechanics. According to this model, the swirling upward air motion in a tornado can be explained even without taking into account the Coriolis force or the vertical convection of warm air. Our results show that for a tornado to appear two factors are necessary: a powerful rotational motion of air in the upper atmosphere and the growth of air pressure from the center to the periphery of an expected region of tornado. The characteristics of air flow in tornado can be found by solving a boundary value problem for a nonlinear parabolic integro-differential equation for an unknown complex-valued temperature. We formulate this problem, propose an algorithm for its numerical solution, and examine its stationary solutions by the stabilization method.

On the Cauchy Problem for Nondegenerate Parabolic Integro-Differential Equations in the Scale of Generalized Hölder Spaces

Parabolic integro-differential nondegenerate Cauchy problem is considered in the scale of Holder spaces of functions whose regularity is defined by a radially O-regularly varying Levy measure. Existence and uniqueness and the estimates of the solution are derived.

A priori $$\varvec{L^{\infty }(L^2)}$$ error estimates for finite element approximations to parabolic integro-differential equations with discontinuous coefficients

Finite element treatment for parabolic integro-differential equations with discontinuous coefficients are presented in this work. Due to low global regularity of the solutions, the error analysis technique of the standard finite element method is difficult to adopt for interface problems. In this paper, convergence of continuous time Galerkin method for the spatially discrete scheme and backward difference scheme in time direction are discussed in $$L^{\infty }(L^2)$$ norm for fitted finite element method with straight interface triangles. More precisely, optimal error estimates are derived in $$L^{\infty }(L^2)$$ norm when initial data $$u_0\in H^3\cap H_0^1(\Omega )$$ . The interface is assumed to be smooth for our purpose.

Two-Grid Methods for a New Mixed Finite Element Approximation of Semilinear Parabolic Integro-Differential Equations

In this paper, we present a two-grid scheme for a semilinear parabolic integro-differential equation using a new mixed finite element method. The gradient for the method belongs to the square integrable space instead of the classical H(div; Ω) space. The velocity and the pressure are approximated by the P02–P1 pair which satisfies the inf-sup condition. Firstly, we solve an original nonlinear problem on the coarse grid in our two-grid scheme. Then, to linearize the discretized equations, we use Newton iteration on the fine grid twice. It is shown that the algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy h = O(H6|lnH|2). As a result, solving such a large class of nonlinear equations will not be much more difficult than the solution of one linearized equation. Finally, a numerical experiment is provided to verify theoretical results of the two-grid method.

Numerical blow-up analysis of linearly implicit Euler method for nonlinear parabolic integro-differential equations

The linearly implicit Euler method for nonlinear parabolic integro-differential equations (NPIDEs) on bounded domains is consideblue by approximating the local effects implicitly and nonlocal effects explicitly. Based on Nakagawa’s criteria, a suitable adaptive time-stepping strategy is introduced by the discrete energy instead of the infinite norm. With the help of lower discrete energies, the finite blow-up behaviors are replicated for any positive solution. Numerical simulations are carried out to examine the effectiveness of our blowup analysis, which also motivate furthermore to prove that a global numerical solution exists for certain NPIDEs with a weakly singular kernel.

Weak Galerkin finite element method for the parabolic integro-differential equation with weakly singular kernel

A weak Galerkin finite element/backward Euler scheme for the parabolic integro-differential equation with weakly singular kernel is considered. The stability and optimal convergence order estimate of the weak Galerkin finite element scheme in \(L^2\) norm are derived. Numerical experiment verifies our theory finding.

Two-grid economical algorithms for parabolic integro-differential equations with nonlinear memory

In this paper, several two-grid finite element algorithms for solving parabolic integro-differential equations (PIDEs) with nonlinear memory are presented. Analysis of these algorithms is given assuming a fully implicit time discretization. It is shown that these algorithms are as stable as the standard fully discrete finite element algorithm, and can achieve the same accuracy as the standard algorithm if the coarse grid size H and the fine grid size h satisfy H=O(hr−1r). Especially for PIDEs with nonlinear memory defined by a lower order nonlinear operator, our two-grid algorithm can save significant storage and computing time. Numerical experiments are given to confirm the theoretical results.

On the Variable Two-Step IMEX BDF Method for Parabolic Integro-differential Equations with Nonsmooth Initial Data Arising in Finance

In this paper the implicit-explicit (IMEX) two-step backward differentiation formula (BDF2) method with variable step-size, due to the nonsmoothness of the initial data, is developed for solving parabolic partial integro-differential equations (PIDEs), which describe the jump-diffusion option pricing model in finance. It is shown that the variable step-size IMEX BDF2 method is stable for abstract PIDEs under suitable time step restrictions. Based on the time regularity analysis of abstract PIDEs, the consistency error and the global error bounds for the variable step-size IMEX BDF2 method are provided. After time semidiscretization, spatial differential operators are treated by using finite difference methods, and the jump integral is computed using the composite trapezoidal rule. A local mesh refinement strategy is also considered near the strike price because of the nonsmoothness of the payoff function. Numerical results illustrate the effectiveness of the proposed method for European and American options under jump-diffusion models.

Two-grid methods for a new mixed finite element approximation of semilinear parabolic integro-differential equations

Two-grid methods for a new mixed finite element approximation of semilinear parabolic integro-differential equations

A two-grid finite element method for nonlinear parabolic integro-differential equations

ABSTRACTIn this paper, we present a two-grid finite element method (FEM) for a two-dimensional nonlinear parabolic integro-differential equation. We solve a fully nonlinear system on a coarse grid space with a grid size H and derive a rough approximation of the exact solution, and then solve the corresponding linearized problem on a fine grid space with a grid size h. The optimal error estimates in -norm are obtained for spatially the semidiscrete two-grid FEM. Finally, numerical examples are given to testify the efficiency of the method.

A Posteriori Error Analysis of the Crank–Nicolson Finite Element Method for Parabolic Integro-Differential Equations

We study a posteriori error analysis for the space-time discretizations of linear parabolic integro-differential equation in a bounded convex polygonal or polyhedral domain. The piecewise linear finite element spaces are used for the space discretization, whereas the time discretization is based on the Crank–Nicolson method. The Ritz–Volterra reconstruction operator (IMA J Numer Anal 35:341–371, 2015), a generalization of elliptic reconstruction operator (SIAM J Numer Anal 41:1585–1594, 2003), is used in a crucial way to obtain optimal rate of convergence in space. Moreover, a quadratic (in time) space-time reconstruction operator is introduced to establish second order convergence in time. The proposed method uses nested finite element spaces and the standard energy technique to obtain optimal order error estimator in the \(L^{\infty }(L^2)\)-norm. Numerical experiments are performed to validate the optimality of the error estimators.

On the null controllability of parabolic equations with nonlinear memory

ABSTRACT In this paper, we study the global null controllability of nonlinear parabolic integro-differential equations arising in reactor dynamics with internal distributed control. Unlike the known results for nonlinear parabolic equations, controllability is analyzed in the frame of strong solutions. The result implies the nonlinear growth of the system with memory which is null controllable can be higher than that of classical parabolic equations. This phenomenon is caused by the memory effects.

Global Lorentz estimates with variable power to linear parabolic integro-differential equations

We shall establish a global estimate in the framework of the Lorentz spaces for variable power of the fractional derivative of solutions to the Cauchy–Dirichlet problems of linear parabolic integro-differential equations with smooth coefficients m(σ)(t,x,y), σ∈[1,2), in a bounded domain. Here, we assume that some higher order derivatives of m(σ)(t,x,y) in y are merely measurable in the time variable t and satisfy uniformly continuous in the spatial variables x, the boundaries of domains are C1,1, and the variable exponents satisfy log-Hölder continuity.

Parameter identification by optimization method for a pollution problem in porous media

In the present work, we investigate the inverse problem of reconstructing the parameter of an integro-differential parabolic equation, which comes from pollution problems in porous media, when the final observation is given. We use the optimal control framework to establish both the existence and necessary condition of the minimizer for the cost functional. Furthermore, we prove the stability and the local uniqueness of the minimizer. Some numerical results will be presented and discussed.

Two-grid methods for expanded mixed finite element approximations of semi-linear parabolic integro-differential equations

In this paper, we investigate a two grid discretization scheme for semilinear parabolic integro-differential equations by expanded mixed finite element methods. The lowest order Raviart–Thomas mixed finite element method and backward Euler method are used for spatial and temporal discretization respectively. Firstly, expanded mixed Ritz–Volterra projection is defined and the related a priori error estimates are proved. Secondly, a superconvergence property of the pressure variable for the fully discretized scheme is obtained. Thirdly, a two-grid scheme is presented to deal with the nonlinear part of the equation and a rigorous convergence analysis is given. It is shown that when the two mesh sizes satisfy h=H2, the two grid method achieves the same convergence property as the expanded mixed finite element method. Finally, a numerical experiment is implemented to verify theoretical results of the two grid method.