Semilinear Parabolic Problems Research Articles

On maximal attractors for dynamical systems. Application to a semilinear parabolic problem with controlled growth

On maximal attractors for dynamical systems. Application to a semilinear parabolic problem with controlled growth

Adaptation of the composite finite element framework for semilinear parabolic problems

In this article, we discuss one of the subsections of finite element method (FEM), classified as the Composite Finite Element Method, abbreviated as CFE. Dimensionality reduction is the primary benefit of the CFE method as it helps to reduce the complexity for the domain space. The degrees of freedom is more in FEM, while compared to the CFE method. Here, the semilinear parabolic problem in a 2D convex polygonal domain is considered. The analysis of the semidiscrete method for the problem is carried out initially in the CFE framework. Here, the discretization would be carried out for the space co-ordinate. Then, fully discrete problem is taken into account, where both the spatial and time components get discretized. In the fully discrete case, the backward Euler method and the Crank-Nicolson method in the CFE framework is adapted for the semilinear problem. The properties of convergence are derived and the error estimates are examined. It is verified that the order of convergence is preserved. The results obtained from the numerical computations are also provided.

Nonsmooth data error estimates for fully discrete finite element approximations of semilinear parabolic equations in Banach space

Implicit–explicit (IMEX) Euler formula together with finite element methods is proposed to solve numerically rather general semilinear parabolic differential equations with initial data in Banach space Xα, 0<α≤1. The time-space regularity of solutions to this class of equations is first investigated. Stability and error estimates are provided for this fully discrete scheme by discrete semigroup method. In the special case when X is the Hilbert space L2(Ω), these error estimates bridge the gap between the existing results on this scheme for semilinear parabolic problems with initial data in L2(Ω) and in H1(Ω) under general nonlinearity. Numerical experiments for nonlinear Allen–Cahn equations verify and complement our theoretical results.

On the regularity of multipliers and second-order optimality conditions of KKT-type for semilinear parabolic control problems

A class of optimal control problems governed by semilinear parabolic equations with mixed constraints and a box constraint for control variable is considered. We show that if the so-called generalized separation condition is satisfied, then both optimality conditions of KKT-type and regularity of multipliers are fulfilled. Moreover, we show that if the initial value is good enough and boundary ∂Ω has a property of positive geometric density, then multipliers and optimal solutions are Hölder continuous.

Unconditional error analysis of linearized BDF2 mixed virtual element method for semilinear parabolic problems on polygonal meshes

In this paper, we construct, analyze, and numerically validate a class of H(div)-mixed virtual element method for the semilinear parabolic problem in mixed form, in which the parabolic problem is reformulated in terms of the velocity and the pressure of the time-dependent Darcy flow. The Newton linearized method for the nonlinear term is designed to cooperate with the second-order backward differentiation formula of the temporal discretization scheme. This allows each time step to only require the solution of a small and well-structured linear system rather than the solution of a nonlinear system. The linearization improves computational efficiency without decreasing convergence rates. Moreover, the “Fortin” operator and its approximation properties are applied to derive an optimal error estimates O(hk+1+τ2) for the virtual element solution of the velocity and the pressure. Finally, its remarkable performance is illustrated by several numerical examples that also validate the theoretical rates of convergence.

Quenching for discretizations of a semilinear parabolic equation with nonlinear boundary outflux

In this paper, we study numerical approximations of a semilinear parabolic problem in one-dimension, of which the nonlinearity appears both in source term and in Neumann boundary condition. By a semidiscretization using finite difference method, we obtain a system of ordinary differential equations which is an approximation of the original problem. We obtain some conditions under which the positive solution of our system quenches in a finite time and estimate its semidiscrete quenching time. Convergence of the numerical quenching time to the theoretical one is established. Next, we show that the quenching rate of the numerical scheme is different from the continuous one. Finally, we give some numerical results to illustrate our analysis.

Elliptic regularization of some semilinear parabolic free boundary problems

Elliptic regularization of some semilinear parabolic free boundary problems

Linearized transformed [formula omitted] finite element methods for semi-linear time-fractional parabolic problems

A linearized Galerkin finite element method is presented for numerically solving the semi-linear time-fractional parabolic problems, whose solutions always display a initial weak singularity. The transformed L1 scheme based on a change of variable is used to approximate Caputo derivatives and the finite element approximations to the spatial variables. By the temporal-spatial error splitting argument, unconditionally optimal error estimates of the proposed schemes are proved. Finally, several numerical experiments are given to demonstrate our theoretical results.

Asymptotic approximations for semilinear parabolic convection-dominated transport problems in thin graph-like networks

We consider time-dependent convection-diffusion problems with high Péclet number of order O(ε−1) in thin three-dimensional graph-like networks consisting of cylinders that are interconnected by small domains (nodes) with diameters of order O(ε). On the lateral surfaces of the thin cylinders and the boundaries of the nodes we account for solution-dependent inhomogeneous Robin boundary conditions which render the associated initial-boundary problem to be nonlinear. The strength of the inhomogeneity is controlled by an intensity factor of order O(εα), α∈R. The asymptotic behavior of the solution is studied as ε→0, i.e., when the diffusion coefficients are eliminated and the thin three-dimensional network is shrunk into a graph. Three qualitatively different cases are discovered in the asymptotic behavior of the solution depending on the value of the intensity parameter α: α=1, α>1, and α<1. We construct the asymptotic approximation of the solution, which provides us with the hyperbolic limit model for ε→0 for the first two cases, and prove the corresponding uniform pointwise estimates and energy estimates. As the main result, we derive uniform pointwise estimates for the difference between the solutions of the convection-diffusion problem and the zero-order approximation that includes the solution of the corresponding hyperbolic limit problem.

Discontinuous Galerkin Time Stepping for Semilinear Parabolic Problems with Time Constant Delay

Discontinuous Galerkin Time Stepping for Semilinear Parabolic Problems with Time Constant Delay

A two-grid immersed finite element method with the Crank-Nicolson time scheme for semilinear parabolic interface problems

In this paper, we propose and analyze a new two-grid partially penalized immersed finite element method for solving the semilinear parabolic interface problems with meshes independent of the coefficient discontinuity. Based on the corresponding time-discrete system, we can unconditionally derive the optimal error estimates in both the L2 norm and semi-H1 norm, while previous works always require the coupling condition of time step and space size (e.g. condition τ=O(H)). Then, we design a two-grid algorithm based on Newton iteration to deal with nonlinear source term. It is shown, both theoretically and numerically, that the algorithm can achieve asymptotically optimal approximation in L2 norm (or semi-H1 norm) when the mesh size satisfies H=O(h3/2) (or H=O(h3)).

An exponential integration generalized multiscale finite element method for parabolic problems

We consider linear and semilinear parabolic problems posed in high-contrast multiscale media in two dimensions. The presence of high-contrast multiscale media adversely affects the accuracy, stability, and overall efficiency of numerical approximations such as finite elements in space combined with some time integrator. In many cases, implementing time discretizations such as finite differences or exponential integrators may be impractical because each time iteration needs the computation of matrix operators involving very large and ill-conditioned sparse matrices. Here, we propose an efficient Generalized Multiscale Finite Element Method (GMsFEM) that is robust against the high-contrast diffusion coefficient. We combine GMsFEM with exponential integration in time to obtain a good approximation of the final time solution. Our approach is efficient and practical because it computes matrix functions of small matrices given by the GMsFEM method. We present representative numerical experiments that show the advantages of combining exponential integration and GMsFEM approximations. The constructions and methods developed here can be easily adapted to three-dimensional domains.

Semilinear periodic parabolic problem with discontinuous coefficients: Mathematical analysis and numerical simulation

We develop a new technique to mathematically analyze and numerically simulate the weak periodic solution to a class of semilinear periodic parabolic equations with discontinuous coefficients. We reformulate our problem into a minimization problem via a least-squares cost function. By using variational calculus theory, we establish the existence of an optimal solution and based on the Lagrangian method, we calculate the derivative of our cost function. To illustrate the validity and efficiency of our proposed method, we present some numerical examples with different periods of time and diverse choices of discontinuous coefficients.

Uniform attractors in sup-norm for semi linear parabolic problem and application to the robust stability theory

In this paper we establish the existence of the uniform attractor for a semi linear parabolic problem with bounded non autonomous disturbances in the phase space of continuous functions. We applied obtained results to prove the asymptotic gain property with respect to the global attractor of the undisturbed system.

A Class of Semilinear Parabolic Problems and Analytic Semigroups

(1) Background: This paper is devoted to the study of a class of semilinear initial boundary value problems of parabolic type. (2) Methods: We make use of fractional powers of analytic semigroups and the interpolation theory of compact linear operators due to Lions–Peetre. (3) Results: We give a functional analytic proof of the C2 compactness of a bounded regular solution orbit for semilinear parabolic problems with Dirichlet, Neumann and Robin boundary conditions. (4) Conclusions: As an application, we study the dynamics of a population inhabiting a strongly heterogeneous environment that is modeled by a class of diffusive logistic equations with Dirichlet and Neumann boundary conditions.

Error Estimates for the Numerical Approximation of Unregularized Sparse Parabolic Control Problems

Abstract We study the numerical approximation of a control problem governed by a semilinear parabolic problem, where the usual Tikhonov regularization term in the cost functional is replaced by a non-differentiable sparsity-promoting term.

Nonautonomous semilinear Wentzell problems in fractal domains

We study a nonautonomous semilinear parabolic problem with a dynamical boundary condition in an irregular domain with fractal boundary. Local existence, uniqueness and regularity results for the mild solution are established via a family of evolution operators. A sufficient condition on the initial datum for global existence is given.

Optimal convergence analysis of a linearized second-order BDF-PPIFE method for semi-linear parabolic interface problems

The article proposes and analyzes the optimal error estimates of a second-order backward difference formula (BDF2) numerical scheme for the semi-linear parabolic interface problems. The partially penalized immersed finite element (PPIFE) methods are used for the spatial discretization to resolve discontinuity of the diffusion coefficient across the interface. The classical extrapolation method is adopted to treat the nonlinear term, which effectively avoids the complicated numerical calculation of the nonlinearity. Our error analysis is based on the corresponding time-discrete system, which neatly splits the error into two parts: the temporal discretization error and the spatial discretization error. Since the spatial discretization error is independent of time step size τ, we can unconditionally derive the optimal error estimates in both L2 norm and semi-H1 norm, while previous works always require the coupling condition of time step and space size. Numerical experiments are given to confirm the theoretical analysis.

Global attractors of mild solutions semiflow for semilinear parabolic equation without uniqueness

In this paper, we establish the existence of global attractor for a semilinear parabolic problem with non-Lipschitz nonlinear term in the phase space of continuous functions.

Blow-Up of the Solution for a Semilinear Parabolic Problem with a Mixed Source

A semilinear parabolic equation with the Dirichlet boundary condition is examined. The reaction source is a mixed nonlinear function. This paper investigates the existence and uniqueness of a solution. A sufficient condition for the solution being blow-up in finite time is found.