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ABSTRACT In this paper, a posteriori error estimates for the fully discrete approximation for linear and semilinear parabolic equations are established. The exponential midpoint method is used for the temporal discretization, and the spatial discretization takes the standard linear elements. By introducing a continuous piecewise quadratic reconstruction, the error of the temporal discretization is derived. The error of the spatial discretization is evaluated by the interpolation estimates. Using the energy technique, we derive residual‐based a posteriori lower and upper bounds for the fully discrete approximation. Our theoretical results are supported by extensive numerical experiments. [ABSTRACT FROM AUTHOR]

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: ∂u∂t+Δp(x)2u+g(x,t,u,∇u)=f(x,t)inQ:=Ω×(0,T),$$ \frac{\partial u}{\partial t}+{\Delta}_{p(x)}^2u+g\left(x,t,u,\nabla u\right)=f\left(x,t\right)\kern3.0235pt \mathrm{in}\kern0.30em \mathcal{Q}:= \Omega \times \left(0,T\right), $$where Ω⊂ℝn(N≥2)$$ \Omega \subset {\mathrm{\mathbb{R}}}^n\kern3.0235pt \left(N\ge 2\right) $$ 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 L+S+C$$ \mathcal{L}+\mathcal{S}+\mathcal{C} $$, in the framework space Lp−(0,T;V)$$ {L}^{p-}\left(0,T;\mathcal{V}\right) $$. [ABSTRACT FROM AUTHOR]

This paper mainly seeks to contribute to the study of a quasilinear parabolic reaction–diffusion system of arbitrary order with initial conditions. Using potential analysis techniques, we establish the existence of continuous positive solutions. The originality lies in the fact that our system is of arbitrary order, which represents a significant advancement as our main result extends and generalizes previous findings.

This article deals with the spectral approximation of an optimal control problem governed by a parabolic partial differential equation (PDE) with an L2$$ {L}^2 $$‐norm control constraint. The investigations employ the space−time spectral method, which is, more precisely, a dual Petrov‐Galerkin spectral method in time and a spectral method in space to discrete the continuous system. As a global method, it uses the global polynomials as the trial functions for discretization of PDEs. After obtaining the optimality condition of the continuous system and that of its spectral discrete surrogate, we establish a priori and a posteriori error estimates for the spectral approximation in detail. Three numerical examples in different spatial dimensions then confirm the theoretical results and also show the efficiency as well as a good precision of the adopted space−time spectral method. [ABSTRACT FROM AUTHOR]

The aim of this work is the numerical homogenization of a parabolic problem with several time and spatial scales using the heterogeneous multiscale method. We replace the actual cell problem with an alternate one, using Dirichlet boundary and initial values instead of periodic boundary and time conditions. Further, we give a detailed a priori error analysis of the fully discretized, i.e., in space and time for both the macroscopic and the cell problem, method. Numerical experiments illustrate the theoretical convergence rates. [ABSTRACT FROM AUTHOR]

In this article we present the nonconforming spectral element method (NSEM) for linear second-order ordinary differential equations with boundary conditions having analytic solutions, and the interface problem. A fully discrete spectral element method for one-dimensional parabolic problems with smooth solutions and parabolic interface problem also have been considered. The stability and error estimates of NSEM for a two-point boundary value problem with analytic solutions are derived. The numerical formulation is described and it is essentially a least-squares formulation and spectral elements are nonconforming. The normal equations which arise in the minimization of the functional are solved using preconditioned conjugate gradient method without storing the stiffness matrix and load vector. The method is exponentially accurate. In the fully discrete formulation for the parabolic problems, Crank–Nicolson scheme is used in time variable and higher-order spectral elements are used in spatial variable. This method is second-order accurate in time and exponential accurate in spatial variable. Various numerical examples are presented to show the accuracy of the method. Finally, we review the nonconforming spectral element method for various problems in higher dimensions. [ABSTRACT FROM AUTHOR]

In this note we consider a parabolic evolution equation in a polygonal space-time cylinder. We show, that the elliptic part is given by a m-accretive mapping from Lq(Ω) → Lq(Ω). Therefore we can apply the theory of nonlinear semigroups in Banach spaces in order to get regularity results in time and space. The second part of the paper deals with the numerical solution of the problem. It is dedicated to the application of the space-time discontinuous Galerkin method (STDGM). It means that both in space as well as in time discontinuous piecewise polynomial approximations of the solution are used. We concentrate to the theoretical analysis of the error estimation. [ABSTRACT FROM AUTHOR]

This paper studies the existence of nonnegative solutions for the following parabolic problem { L ω u + ∂ u ∂ t = 0 , in R d × (0 , T) , u (x , 0) = u 0 (x) , in R d , where T>0, L ω is a selfadjoint operator associated with a regular Dirichlet form Q, the initial value u 0 ∈ L 2 (R d) , u 0 ≥ 0 is a Borel measurable function. [ABSTRACT FROM AUTHOR]

We introduce and analyze a very simple but efficient postprocessing technique for improving the accuracy of the continuous Galerkin (CG) time stepping method for nonlinear first-order initial value problems with smooth and singular solutions. The key idea of the postprocessing technique is to add a higher order Lobatto polynomial of degree k + 1 to the CG solution of degree k. We first establish global superconvergent error bounds for the postprocessed CG approximations over arbitrary time partitions, and as a by-product, it is shown that the convergence rates of the L 2 -, H 1 - and L ∞ -estimates for the CG method with quasi-uniform meshes for smooth solutions are improved by one order. Moreover, for solutions with initial singularities, we prove that the optimal global error estimates and nodal superconvergent estimate can be obtained for the CG method with graded meshes, and after postprocessing, the convergence rates of the L 2 -, H 1 - and L ∞ -estimates are also improved by one order. As an application, we apply the superconvergent postprocessing technique to the CG time discretization of nonlinear parabolic equations. Numerical examples are presented to verify the theoretical results. [ABSTRACT FROM AUTHOR]

This study aims to develop a robust numerical algorithm for solving parabolic partial differential equations (PDEs) arising in the domain of financial mathematics. The proposed approach leverages the finite difference method (FDM) to discretize the temporal and spatial domains of the problem. To approximate the unknown solution, we employ a polynomial interpolation technique, ensuring high accuracy and stability in the numerical solution. The effectiveness and efficiency of our method are demonstrated through comprehensive numerical experiments, showcasing its potential for practical applications in financial modeling. [ABSTRACT FROM AUTHOR]

We consider some problems governed by the sum of a Laplacian and of a fractional Laplacian in presence of so–called (α , β) -Neumann conditions in an essentially linear context. Indeed, first we show an existence result for asymptotically linear elliptic problems, and then we establish some qualitative properties for solutions of the associated homogeneous parabolic problem, extending to this setting related results in the local case. [ABSTRACT FROM AUTHOR]

In this paper, we study and construct a higher order numerical algorithm for singularly perturbed differential-difference convection–diffusion equation with retarded term appearing in computational biological science. The solution of the considered class of problems may exhibit a boundary layer due to the presence of the perturbation parameter and retarded term. We have discussed the analytical behaviour of the exact solution and its partial derivatives. We have used Taylor's series expansion to approximate the advance and delay terms of the model problem and then the problem is discretized using Crank–Nicolson's method on equidistant mesh in the time direction and modified cubic B-spline basis functions on generalized Shishkin mesh in the spatial direction. We have also shown that the developed algorithm is unconditionally stable. The proposed numerical algorithm is proved to be ε -uniformly convergent of order four in the spatial direction up to a logarithmic factor and second-order convergent in the time direction. To demonstrate the accuracy and to validate the theoretical results of the proposed numerical algorithm, we have presented three numerical experiments. We have also compared our method with existing schemes in the literature to prove the accuracy of the proposed numerical algorithm. [ABSTRACT FROM AUTHOR]

In this paper we establish compactness results of multiscale and very weak multiscale type for sequences bounded in $L^2(0,T;H_0^1(\Omega))$, fulfilling a certain condition. We apply the results in the homogenization of the parabolic partial differential equation $\varepsilon^p\partial_tu_{\varepsilon}(x,t) -\nabla\cdot( a( x\varepsilon^{-1} ,x\varepsilon^{-2},t\varepsilon^{-q},t\varepsilon^{-r}) \nabla u_{\varepsilon}(x,t) ) = f(x,t) $, where $0

We prove an entropy solutions for a class of strongly nonlinear parabolic problems with lower order terms in Musielak-Sobolev spaces, without sign condition on the nonlinearities and with measure data. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the model problem: ∂u∂t−Δpu=finQu=0on∂Ω×(0,T)u(x,0)=u0inΩ. The main purpose of this work is to prove the existence and a limiting regularity result for the solution u of the above problem having right‐hand side f∈Lβ(0,T;LLogαL(Ω)). In particular, we will consider the cases: (i)If p≥2,β=1, and α>N−1N, then u∈Lp−1(0,T,W01,q‾(Ω))q‾=N(p−1)N−1.(ii)If 2−1NN−mN, then u∈Lp−1(0,T,W01,m∗(p−1)(Ω)). [ABSTRACT FROM AUTHOR]

This work is devoted to the development of a Galerkin-type approximation of the solution of parabolic reaction-diffusion problems, utilizing cubic B-Spline functions and a finite difference scheme. An error estimate for the semi discrete weak Galerkin scheme is established. A Von Neumann stability study of the proposed fully discrete Crank Nicolson scheme is also performed. In addition, examples are used to validate the proposed approximation. The numerical results produced demonstrate the procedure’s efficacy and are in good agreement with the exact solution. [ABSTRACT FROM AUTHOR]

In this study, we present new sharper constructive a priori error estimates for a full-discrete numerical solution of the heat equation that refines our previous work. The full discretization is given using the finite element method in space and linear interpolation in time, similar to that in the previous work. In particular, we adopt an approach based on the effective use of the properties both for exact and discretized time-periodic solutions to establish the error estimates simpler and sharper than the previous work. Furthermore, we derive some time-stepwise error estimates in the space direction. Finally, we present several numerical examples that confirm the actual refinements of the convergence. [ABSTRACT FROM AUTHOR]

We study an inverse parabolic problem of identifying two source terms in heat equation with dynamic boundary conditions from a final time overdetermination data. Using a weak solution approach by Hasanov, the associated cost functional is analyzed, especially a gradient formula of the functional is proved and given in terms of the solution of an adjoint problem. Next, the existence and uniqueness of a quasi‐solution are also investigated. Finally, the numerical reconstruction of some heat sources in a 1‐D equation is presented to show the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

We consider a family of semilinear parabolic problems with nonlinear boundary conditions $$\displaylines{ u_t(x,t)=\Delta u(x,t) -au(x,t) + f(u(x,t)),\quad x \in \Omega_\epsilon,\; t>0\,,\cr \frac{\partial u}{\partial N}(x,t)=g(u(x,t)), \quad x \in \partial\Omega_\epsilon,\; t>0\,, }$$ where $\Omega_0 \subset \mathbb{R}^n$ is a smooth (at least $\mathcal{C}^2$) domain, $\Omega_{\epsilon} = h_{\epsilon}(\Omega_0)$ and $h_{\epsilon}$ is a family of diffeomorphisms converging to the identity in the $\mathcal{C}^1$-norm. Assuming suitable regularity and dissipative conditions for the nonlinearites, we show that the problem is well posed for $\epsilon>0$ sufficiently small in a suitable scale of fractional spaces, the associated semigroup has a global attractor $\mathcal{A}_{\epsilon}$ and the family $\{\mathcal{A}_{\epsilon}\}$ is continuous at $\epsilon = 0$.

This paper deals with the problem of determining an unknown source and an unknown initial condition in a abstract final value parabolic problem. This problem is ill-posed in the sense that the solutions do not depend continuously on the data. To solve the considered problem a modified Tikhonov regularization method is proposed. Using this method regularized solutions are constructed and under boundary conditions assumptions, convergence estimates between the exact solutions and their regularized approximations are obtained. Moreover numerical results are presented to illustrate the accuracy and efficiency of the proposed method.

This paper is concerned with the existence and blow-up of solutions to the following linear parabolic equation: ut−Δu+u = 0 in Ω×(0, T), under nonlinear boundary condition in a bounded domain Ω ⊂ Rn, n ≥ 1, with smooth boundary. We obtain a threshold result for the global existence of solutions, next we shall prove the existence time T of solution is finite when the initial energy satisfies certain condition. [ABSTRACT FROM AUTHOR]

This work is devoted to an inverse problem of identifying a source term depending on both spatial and time variables in a parabolic equation from single Cauchy data on a part of the boundary. A Crank–Nicolson Galerkin method is applied to the least squares functional with a quadratic stabilizing penalty term. The convergence of finite dimensional regularized approximations to the sought source as measurement noise levels and mesh sizes approach zero with an appropriate regularization parameter is proved. Moreover, under a suitable source condition, an error bound and a corresponding convergence rate are proved. Finally, several numerical experiments are presented to illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper we establish compactness results of multiscale and very weak multiscale type for sequences bounded in.... The homogenization result reveals two special phenomena, namely that the homogenized problem is elliptic and that the matching for which the local problem is parabolic is shifted by p, compared to the standard matching that gives rise to local parabolic problems. [ABSTRACT FROM AUTHOR]

We consider a general inhomogeneous parabolic initial-boundary value problem for a $ 2b $-parabolic differential equation given in a finite multidimensional cylinder. We investigate the solvability of this problem in some generalized anisotropic Sobolev spaces. They are parametrized with a pair of positive numbers $ s $ and $ s/(2b) $ and with a function $ \varphi:[1,\infty)\to(0,\infty) $ that varies slowly at infinity. The function parameter $ \varphi $ characterizes subordinate regularity of distributions with respect to the power regularity given by the number parameters. We prove that the operator corresponding to this problem is an isomorphism on appropriate pairs of these spaces. As an application, we give a theorem on the local regularity of the generalized solution to the problem. We also obtain sharp sufficient conditions under which chosen generalized derivatives of this solution are continuous on a given set. [ABSTRACT FROM AUTHOR]

In this article, we study the existence and uniqueness of nonnegative solutions for the following parabolic problem { H t ω u + ∂ u ∂ t = 0 , in X × (0 , T) , u (x , 0) = u 0 (x) , in X , where T > 0, X is a locally compact separable metric space, Ht is a operator (not necessarily selfadjoint) associated with a regular (not necessarily symmetric)local Dirichlet form E t , the initial value u 0 ∈ L 2 (X , ω 2 d m) and m is a positive Radon measure on Borel subset U of X such that m(U) > 0. Our technical approach for uniqueness is based on Täcklind type uniqueness theorem. [ABSTRACT FROM AUTHOR]

The multi-scale analysis method is presented for the heat transfer problems of periodical composite structures under irradiation heating. This kind of physical problem can be described as a class of parabolic equations with small periodical oscillating coefficients equipped with the mixed boundary conditions of both Dirichlet and Neumann type. It needs more expensive cost in both computer's memory and CPU time to solve these problems by the usual finite difference method or finite element method, since it is necessary to partition the considering geometrical region finer to capture the heat conduction behavior at small scale. To reduce the computational complexity, an approximate solution is derived by the asymptotic expansion technique. The convergence of asymptotic solution to exact solution is technically proved as the micro-scale parameter tends to zero. Based on the asymptotic expansion formula, a second-order two-scale finite element method is proposed to fully solve this problem in practice. Numerical results show convergent behavior of the proposed method. [ABSTRACT FROM AUTHOR]

We consider a family of semilinear parabolic problems with non- linear boundary conditions... [ABSTRACT FROM AUTHOR]

In this paper, we propose a space-time least-squares isogeometric method to solve parabolic evolution problems, well suited for high-degree smooth splines in the space-time domain. We focus on the linear solver and its computational efficiency: thanks to the proposed formulation and to the tensor-product construction of space-time splines, we can design a preconditioner whose application requires the solution of a Sylvester-like equation, which is performed efficiently by the fast diagonalization method. The preconditioner is robust w.r.t. spline degree and mesh size. The computational time required for its application, for a serial execution, is almost proportional to the number of degrees-of-freedom and independent of the polynomial degree. The proposed approach is also well-suited for parallelization. [ABSTRACT FROM AUTHOR]

Summary: The heat equation is solved by using a finite volume discretization in a domain that consists of a two‐dimensional central node and several one‐dimensional outgoing branches. Several interface connection options to match the submodels set on the node and on the branches, with or without continuity, are looked at. For each of them, a monolithic scheme is defined, and existence and uniqueness of the solution is proved. New schemes are deduced, which are obtained through domain decomposition methods in the form of interface systems, with one or two unknowns per interface. A comparative systematic study is carried out from an algebraic and numerical point of view according to the interface conditions: Dirichlet, Neumann, or Robin. An efficient diagonal preconditioning is proposed. [ABSTRACT FROM AUTHOR]

In this paper, the blow-up solutions for a parabolic problem, defined in a bounded domain, are studied. Namely, we consider the upper blow-up rate estimate for heat equation with a nonlinear Neumann boundary condition defined on a ball in . [ABSTRACT FROM AUTHOR]

In this article, we study numerical approximation for a class of singularly perturbed parabolic (SPP) convection‐diffusion turning point problems. The considered SPP problem exhibits a parabolic boundary layer in the neighborhood of one of the sides of the domain. Some a priori bounds are given on the exact solution and its derivatives, which are necessary for the error analysis. A numerical scheme comprising of implicit finite difference method for time discretization on a uniform mesh and a hybrid scheme for spatial discretization on a generalized Shishkin mesh is proposed. Then Richardson extrapolation method is applied to increase the order of convergence in time direction. The resulting scheme has second‐order convergence up to a logarithmic factor in space and second‐order convergence in time. Numerical experiments are conducted to demonstrate the theoretical results and the comparative study is done with the existing schemes in literature to show better accuracy of the proposed schemes. [ABSTRACT FROM AUTHOR]

We show maximal regularity results concerning parabolic systems with dynamic boundary conditions and a diffusion theorem on the boundary in the framework of L p spaces, 1 < p < ∞ . Analyticity results can be derived for the semigroups generated by suitable classes of uniformly elliptic operators with general Wentzell boundary conditions having diffusion terms on the boundary. [ABSTRACT FROM AUTHOR]

Optical systems with two-dimensional feedback demonstrate wide possibilities for studying the nucleation and development processes of dissipative structures. Feedback allows to influence the dynamics of the optical system by controlling the transformation of spatial variables performed by prisms, lenses, dynamic holograms and other devices. A nonlinear interferometer with a mirror image of a field in two-dimensional feedback is one of the simplest optical systems in which is realized the nonlocal nature of light fields. A mathematical model of optical systems with two-dimensional feedback is a nonlinear parabolic equation with rotation transformation of a spatial variable and periodicity conditions on a circle. Such problems are investigated: bifurcation of the traveling wave type stationary structures, how the form of the solution changes as the diffusion coefficient decreases, dynamics of the solutions stability when the bifurcation parameter leaves the critical value. For the first time as a parameter bifurcation was taken of diffusion coefficient. The method of central manifolds and the Galerkins method are used in this paper. The method of central manifolds and the Galerkins method are used in this paper. The method of central manifolds allows to prove a theorem on the existence and form of the traveling wave type solution neighborhood of the bifurcation value. The first traveling wave born as a result of the Andronov -Hopf bifurcation in the transition of the bifurcation parameter through the сritical value. According to the central manifold theorem, the first traveling wave is born orbitally stable. Since the above theorem gives the opportunity to explore solutions are born only in the vicinity of the critical values of the bifurcation parameter, the decision to study the dynamics of traveling waves of change during the withdrawal of the bifurcation parameter in the supercritical region, the formalism of the Galerkin method was used. In accordance with the method of the central manifold is made Galerkins approximation of the problem solution. As the bifurcation parameter decreases and its transition through the critical value, the zero solution of the problem loses stability in an oscillatory manner. As a result, a periodic solution of the traveling wave type branches off from the zero solution. This wave is born orbitally stable. With further reduction of the parameter and its passage through the next critical value from the zero solution, the second solution of the traveling wave type is produced as a result of the Andronov -Hopf bifurcation. This wave is born unstable with an instability index of two. Numerical calculations have shown that the application of the Galerkins method leads to correct results. The results obtained are in good agreement with the results obtained by other authors and can be used to establish experiments on the study of phenomena in optical systems with feedback.

The nonconforming virtual element method for the parabolic problem is developed in this paper, where the semi-discrete and fully discrete formulations are presented. In order to analyze the convergence of the method, we construct a projection operator in the sense of the energy norm and give the corresponding error estimates in the L 2 norm and broken H 1 semi-norm. With the help of the energy projection operator, we prove the optimal convergence of the nonconforming virtual element method in the semi-discrete and fully discrete formulations, respectively. Finally, we investigate the convergence of the nonconforming virtual element method by some numerical examples, which verify the theoretical results. [ABSTRACT FROM AUTHOR]

We provide new insights into the a priori theory for a time‐stepping scheme based on least‐squares finite element methods for parabolic first‐order systems. The elliptic part of the problem is of general reaction‐convection‐diffusion type. The new ingredient in the analysis is an elliptic projection operator defined via a nonsymmetric bilinear form, although the main bilinear form corresponding to the least‐squares functional is symmetric. This new operator allows to prove optimal error estimates in the natural norm associated to the problem and, under additional regularity assumptions, in the L2 norm. Numerical experiments are presented which confirm our theoretical findings. [ABSTRACT FROM AUTHOR]

We study a space-time coupled least-squares spectral element method for parabolic initial boundary value problems using parallel computers. The method is spectrally accurate in both space and time. The spectral element functions are polynomials of degree q in the time variable and of degree p in each of the space variables and are non-conforming in both space and time. There is no need therefore to consider C0 element expansions at the inter element boundaries and consequently it is unnecessary to express the equation as an equivalent set of first-order equations by introducing an additional independent variable. Instead, we consider the jump in the function and its derivatives across inter-element boundaries in certain Sobolev norms. The normal equations obtained from the least-squares formulation can be solved by a preconditioned conjugate gradient method (PCGM) in matrix-free form. The method is implemented on a parallel computer using message-passing-interface (MPI). We take q proportional to p2 in the p-version of the method, whereas p = 2 q + 1 in the h-version. If the solution belongs to a certain Gevrey space, then the error decays exponentially in p in the p-version of the method and the error is O (h 2 q − 1) in the h-version. Further, the number of iterations, needed to obtain the approximate solution using PCGM, is O ( ( ln p) 2 / h) per time step. These estimates are confirmed by the computational results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the well posedness and homogenization for a multiscale parabolic problem in a cylinder Q of RN. A rapidly oscillating non-smooth interface inside Q separates the cylinder in two heterogeneous connected components. The interface has a periodic microstructure, and it is situated in a small neighborhood of a hyperplane which separates the two components of Q. The problem models a time-dependent heat transfer in two heterogeneous conducting materials with an imperfect contact between them. At the interface, we suppose that the flux is continuous and that the jump of the solution is proportional to the flux. On the exterior boundary, homogeneous Dirichlet boundary conditions are prescribed. We also derive a corrector result showing the accuracy of our approximation in the energy norm. [ABSTRACT FROM AUTHOR]