Parabolic Problems Research Articles

Existence and regularity results of parabolic problems with convection term and singular nonlinearity

In this work, we investigate the influence of the convection term and the singular lower order term on the existence and regularity of solutions to the following parabolic problem: \begin{cases}\frac{\partial u}{\partial t}-\operatorname{div}(M(x,t)\nabla u) =-\operatorname{div}(uE(x,t))+\frac{f}{u^{\theta}}&\text{in }\Omega\times(0,T),\\ u(x,t)=0&\text{on }\partial \Omega\times (0,T),\\ u(x,0)=u_{0}(x)&\text{in }\Omega,\end{cases} where \theta>0 , \Omega\subset \mathbb{R}^{N}\ (N>2) is a bounded smooth domain with 0\in \Omega , and f\in L^{m}(\Omega\times (0,T)) with m\geq 1 is a non-negative function. The function u_{0} is a non-negative function that belongs to the space L^{\infty}(\Omega) such that \forall \omega\subset\subset \Omega,\ \exists c_{\omega}>0,\quad u_{0}\geq c_{\omega}\text{ in }\omega. The main idea of this research explains the combined impact of the convection term and the singular lower order term on the existence and regularity of a solution to the above problem.

Parabolic Boundary-Value Problems of Mathematical Physics in a Piecewise Homogeneous Wedge-Shaped Hollow Cylinder

Parabolic Boundary-Value Problems of Mathematical Physics in a Piecewise Homogeneous Wedge-Shaped Hollow Cylinder

A Fisher-type evolution equation with applications in image processing

The aim of this paper is to present some applications of the Neumann Laplacian in image processing, along with the necessary mathematical background. We prove weak and strong versions of the maximum principle for weak solutions of elliptic and parabolic problems and apply them to a Fisher K.P.P.-type equation. The original contribution lies in the application of this equation in image processing, where various diffusion-like effects can be achieved. Additionally, a review of the basics of linear and nonlinear PDEs with Neumann boundary conditions is provided, along with updated bibliography and recent qualitative results. There are also some new theoretical results developed in this work.

Theoretical analysis and numerical simulation of a weak periodic solution for a parabolic problem with nonlinear boundary conditions

The aim of this work is to develop a numerical tool for computing the weak periodic solution for a class of parabolic equations with nonlinear boundary conditions. We formulate our problem as a minimization problem by introducing a least-squares cost function. With the help of the Lagrangian method, we calculate the gradient of the cost function. We build an iterative algorithm to simulate numerically the weak periodic solution to the considered problem. To illustrate our approach, we present some numerical examples.

A parallel-in-time preconditioner for Crank–Nicolson discretization of a parabolic optimal control problem

In this paper, a fast solver is studied for saddle point system arising from a second-order Crank–Nicolson discretization of an initial-valued parabolic PDE constrained optimal control problem, which is indefinite and ill-conditioned. Different from the saddle point system arising from the first-order Euler discretization, the saddle point system arising from Crank–Nicolson discretization has a dense and non-symmetric Schur complement, which brings challenges to fast solver designing. To remedy this, a novel symmetrization technique based on the block Toeplitz structure is applied to the saddle point system so that the new Schur complement is symmetric definite and the well-known matching-Schur-complement (MSC) preconditioner is applicable to the new Schur complement. Nevertheless, the new Schur complement is still a dense matrix and the inversion of the corresponding MSC preconditioner is not parallel-in-time (PinT) and thus time consuming. For this concern, a modified MSC preconditioner for the new Schur complement system. Our new preconditioner can be implemented in a fast and PinT way via a temporal diagonalization technique. Theoretically, the eigenvalues of the preconditioned matrix by our new preconditioner are proven to be lower and upper bounded by positive constants independent of matrix size and the regularization parameter. With such spectrum, the preconditioned conjugate gradient (PCG) solver for the Schur complement system is proven to have a convergence rate independent of matrix size and regularization parameter. To the best of my knowledge, it is the first time to have an iterative solver with problem-independent convergence rate for the saddle point system arising from Crank–Nicolson discretization of the optimal control problem. Numerical results are reported to show that the performance of the proposed preconditioner.

Time-Stepping Error Estimates of Linearized Grünwald–Letnikov Difference Schemes for Strongly Nonlinear Time-Fractional Parabolic Problems

A fully discrete scheme is proposed for numerically solving the strongly nonlinear time-fractional parabolic problems. Time discretization is achieved by using the Grünwald–Letnikov (G–L) method and some linearized techniques, and spatial discretization is achieved by using the standard second-order central difference scheme. Through a Grönwall-type inequality and some complementary discrete kernels, the optimal time-stepping error estimates of the proposed scheme are obtained. Finally, several numerical examples are given to confirm the theoretical results.

On the stability of θ-methods for DDEs and PDDEs

In this paper, the stability of θ-methods for delay differential equations is studied based on the test equation y′(t)=−Ay(t)+By(t−τ), where τ is a constant delay and A is a positive definite matrix. It is mainly considered the case where the matrices A and B are not simultaneosly diagonalizable and the concept of field of values is used to prove a sufficient condition for unconditional stability of these methods and another condition which also guarantees their stability, but according to the step size. The results obtained are also simplified for the case where the matrices A and B are simultaneously diagonalizable and compared with other similar works for the general case. Several numerical examples in which the theory discussed here is applied to parabolic problems given by partial delay differential equations with a diffusion term and a delayed term are presented, too.

Error Identities for Parabolic Equations with Monotone Spatial Operators

Abstract The article studies quantitative relations (error identities) that characterize distances between exact solutions of nonlinear evolutionary problems and functions considered as approximations. The restrictions imposed on such a function are minimal and actually come down to the condition that it belongs to the same functional class as the generalized solution of the problem under consideration. Functional identities of this type reflect the most general relations between deviations from exact solutions of parabolic initial boundary value problems and those data that can be observed in a numerical experiment. The identities contain no mesh dependent constants and are valid for any function in the admissible (energy) class regardless of the method by which it was constructed. Therefore, they can serve as basic tools for deriving fully reliable a posteriori estimates of approximation errors as well as for analysis of modeling errors. The corresponding examples are discussed in the paper.

A single mesh approximation for modeling multiphase flow in heterogeneous porous media

The control volume finite element (CVFE) approach is based on the discretization of primary flow and transport unknowns on two different meshes. The element mesh, used to represent the physical properties of the medium, differs from the control volume mesh necessary to ensure a mass conservative solution. The inherent two mesh feature of the CVFE approximation introduces inconsistency in the transport solution due to the averaging of physical and computed quantities between neighboring elements in the mesh. In this work, we present a consistent approach for modeling multiphase flow and transport in heterogeneous porous media. The approach is applied in the CVFE framework by enabling the discretization of primary unknowns on a single mesh. We combine the discretization of an element-wise, discontinuous pressure approximation with a first-order, discontinuous velocity approximation to resolve the elliptic or parabolic flow problem. The effectiveness of the formulation is achieved by exploiting the same finite element mesh as the flow problem, when updating the saturation solution. This direct mapping between the flow and transport mesh is simple yet effective for establishing a consistent solution in addition to circumventing the non-physical mass leakage exhibited in the classical CVFE method. We describe the interface approximations and the discontinuous terms needed for consistent solutions. The method is well suited to model flow in complex geometrical subsurface domains and is shown to be numerically stable while providing locally and globally mass conservative solutions. We apply the approach to several domains with complex geometrical features to emphasize the superiority of the approximation over conventional methods. The analysis shows over two orders of magnitude reduction in solution error compared to the classical CVFE. The new formulation effectively captures accurate transport solutions, even in the presence of varying material properties, without the need for mesh refinement.

An efficient numerical approach for singularly perturbed time delayed parabolic problems with two-parameters

ObjectivesThe main objective of this work is to design an efficient numerical scheme is proposed for solving singularly perturbed time delayed parabolic problems with two parameters.ResultsThe scheme is constructed via the implicit Euler and non-standard finite difference method to approximate the time and space derivatives, respectively. Besides, to enhance the accuracy and order of convergence of the method Richardson extrapolation technique is employed. Intensive numerical experimentation has been done on some model examples. Further, the layer behavior of the solutions is presented using graphs and observed to agree with the existing theories. Finally, error analysis of the scheme is done and observed that the proposed method is parameter uniform convergent with the order of convergence (Δt)2+h2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( (\\Delta t )^2+h^2\\right) $$\\end{document}

Homogenization of partial differential equations with Preisach operators

The current work deals with initial boundary value parabolic problems with Preisach hysteresis whose the density functions are allowed to depend on the variable of space. The model contains nonlinear monotone operators in the diffusion term, arising from an energy. Thanks to the properties of Preisach hysteresis operators and to the sigma-convergence method, we obtain the convergence of themicroscopic solutions to the solution of the homogenized problem. The effective operator is obtained in terms of a solution of a nonlinear corrector equation addressed in the usual sense of distributions, leading in an approximate scheme for the homogenizedcoefficient which is an important step towards the numerical implementation of the results from the homogenization theory beyond the periodic setting.

A fitted parameter convergent finite difference scheme for two-parameter singularly perturbed parabolic differential equation

The objective of this paper is to develop a numerical scheme that is uniform in its parameters for a specific type of time-dependent parabolic problem with two perturbation parameters. The existence of these two parameters in the terms with the highest-order derivatives results in the formation of boundary layer(s) in the solution of such problems. Solving these model problems using classical methods does not yield satisfactory results due to the layer behavior. Therefore, nonstandard finite difference schemes have been developed as a means to obtain numerical solutions for these problems. To develop the scheme, we employ the Crank-Nicolsondiscretization on a uniform time mesh and apply a fitted operator method with a uniform spatial mesh. We have established the stability and convergence of the proposed scheme. The proposed scheme exhibits uniform convergence of second order in the temporal direction and first order in the spatial direction. However, temporal mesh refinements is employed to enhance the order to two in both directions. Model examples are provided to validate the practicality of the proposed numerical scheme.

Well-posedness and parabolicity of the boundary-value problem for systems of partial differential equations

The paper examines two-point boundary value problem for systems of linear partial differential equations. Not every system has a well posed two-point boundary value problem. For example, for equation {$$\displaystyle\frac{\partial u(x_1,x_2,t)}{\partial t}=\displaystyle\frac{\partial u(x_1,x_2,t)}{\partial x_1}+i \frac{\partial u(x_1,x_2,t)}{\partial x_2}$$ there are no boundary conditions of the form $au(x_1,x_1,0)+bu(x_1,x_2,T)=\varphi(x_1,x_2)$, under which this boundary value problem will be of a well posed in the Schwartz spaces. In the work, the conditions for the matrix of the system under which exist well posed boundary-value problem are found in the Schwartz spaces, and the form of these boundary conditions is also indicated. So, for systems with a Hermitian matrix, the boundary value problem with conditions of form $$u(x,0)+u(x,T)=\varphi(x)$$ will always be of a well posed in the Schwartz spaces, as well as in the spaces of functions of finite smoothness of power growth. For systems with one spatial variable, it is proved that well posed boundary value problems with the condition $u(x,0)+bu(x,T)=\varphi(x)$ ith positive b always exist. In addition, parabolic boundary value problems with property of increasing smoothness of the solutions are investigated. Similar results hold for linear partial differential equations. The Helmholtz equation $$\displaystyle\frac{\partial^2 u(x,t)}{\partial t^2}+\Delta u(x,t)=ku(x,t),$$ is considered as an example. It is not well posed according to Petrovsky, but for it exist a boundary value problem that is parabolic. For an equation with one spatial variable, sufficient conditions for the second-order equation in time are given, under which parabolic boundary value problems exist.

Two-grid Method of Expanded Mixed Finite Element Approximations for Parabolic Integro-differential Optimal Control Problems

Two-grid Method of Expanded Mixed Finite Element Approximations for Parabolic Integro-differential Optimal Control Problems

Blow-Up of Solution and Energy Decay for a Quasilinear Parabolic Problem

Blow-Up of Solution and Energy Decay for a Quasilinear Parabolic Problem

Space-time error estimates for approximations of linear parabolic problems with generalized time boundary conditions

Abstract We first give a general error estimate for the nonconforming approximation of a problem for which a Banach–Nečas–Babuška (BNB) inequality holds. This framework covers parabolic problems with general conditions in time (initial value problems as well as periodic problems) under minimal regularity assumptions. We consider approximations by two types of space-time discretizations, both based on a conforming Galerkin method in space. The first one is the Euler $\theta -$scheme. In this case, we show that the BNB inequality is always satisfied, and may require an extra condition on the time step for $\theta \le \frac 1 2$. The second one is the time discontinuous Galerkin method, where the BNB condition holds without any additional condition.

Goal-oriented adaptive space-time finite element methods for regularized parabolic p-Laplace problems

We consider goal-oriented adaptive space-time finite-element discretizations of the regularized parabolic p-Laplace problem on completely unstructured simplicial space-time meshes. The adaptivity is driven by the dual-weighted residual (DWR) method since we are interested in an accurate computation of some possibly nonlinear functionals at the solution. Such functionals represent goals in which engineers are often more interested than the solution itself. The DWR method requires the numerical solution of a linear adjoint problem that provides the sensitivities for the mesh refinement. This can be done by means of the same full space-time finite element discretization as used for the primal non-linear problems. The numerical experiments presented demonstrate that this goal-oriented, full space-time finite element solver efficiently provides accurate numerical results for different functionals.

Mixed Galerkin- Implicit Differences Methods for Solving Coupled Parabolic Boundary Value Problems with Variable Coefficients

Mixed Galerkin- Implicit Differences Methods for Solving Coupled Parabolic Boundary Value Problems with Variable Coefficients

The parabolic thin obstacle problem for the weighted biLaplacian

The parabolic thin obstacle problem for the weighted biLaplacian

Weak solutions of a nonlinear degenerate fourth‐order parabolic equation via the topological degree method

In this paper, we study the existence of weak solutions for the nonlinear parabolic problem associated with the following fourth‐order equation, under Navier boundary conditions: where is a bounded domain. Under some appropriate assumptions, we obtain the existence of a weak solution for the above problem by using the topological degree theory for operators of the type , in the framework space .