Order Parabolic Problems Research Articles

A stabilizer free weak Galerkin method with implicit [formula omitted]-schemes for fourth order parabolic problems

In this study, we solve the fourth-order parabolic problem by combining the implicit θ-schemes in time for θ∈[12,1] with the stabilizer free weak Galerkin (SFWG) method. The semi-discrete and full-discrete numerical schemes are proposed. And specifically, the full-discrete scheme is a first-order backward Euler scheme when θ=1, and a second-order Crank–Nicolson scheme for θ=12. Then, we determine the optimal convergence orders of the error in the H2 and L2 norms after analyzing the well-posedness of the schemes. The theoretical findings are validated by numerical experiments.

An efficient numerical approach for singularly perturbed parabolic convection-diffusion problems with large time-lag

In this paper, an efficient finite difference method is presented for solving singularly perturbed linear second order parabolic problems with large time lag. The comparable numerical model is related to automatically controlled system with spatial diffusion of reactants in the processes. This study focuses on the formation of boundary layer behavior or oscillatory behaviors due to the presence of delay parameters and perturbation parameter. The numerical scheme comprising an exponentially fitted spline based difference scheme on a uniform mesh supported by Crank-Nicolson Method is constructed. It is found that the present method converges with second order accurate in both temporal and spatial variables. The convergence analysis and running time of the program with varied grid sizes are then used to do the efficiency analysis. The proposed scheme accuracy and efficiency are also demonstrated through numerical experiments.

Sharp estimates for homogeneous semigroups in homogeneous spaces. Applications to PDEs and fractional diffusion in ℝN

In this paper, we analyze evolution problems associated to homogenous operators. We show that they have an homogenous associated semigroup of solutions that must satisfy some sharp estimates when acting on homogenous spaces and on the associated fractional power spaces. These sharp estimates are determined by the homogeneity alone. We also consider fractional diffusion problems and Schrödinger type problems as well. We apply these general results to broad classes of PDE problems including heat or higher order parabolic problems and the associated fractional and Schrödinger problems or Stokes equations. These equations are considered in Lebesgue or Morrey spaces.

Primal hybrid finite element method for fourth order parabolic problems

In this article, we discuss a primal hybrid finite element method for a fourth order parabolic initial-boundary value problem. The interelement continuity requirement has been alleviated using Lagrange multipliers on the interelement boundaries. Primal hybrid finite elements are constructed and used in spatial direction and Crank-Nicolson scheme is used in the temporal direction for solving complete discrete scheme. Optimal order estimates for both semidiscrete and complete discrete scheme are derived with the help of a non-orthogonal projection operator. Numerical results are presented in order to validate the theoretical findings.

Parametrix method for the first hitting time of an elliptic diffusion with irregular coefficients

ABSTRACT In this article, we are interested in studying the transition density function of the couple given by the first hitting time of a fixed threshold by a one-dimensional uniformly elliptic diffusion process and the associated stopped process. Our strategy relies on the parametrix technique applied to the related semigroup. It notably allows us to extend standard PDEs results on Green and Poisson functions, see e.g. Garroni and Menaldi [Green functions for second order parabolic integro-differential problems, volume 275 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1992], by establishing the infinite expansion of the corresponding transition density for irregular coefficients, here bounded measurable drift coefficient and bounded η-Hölder-continuous diffusion coefficient. As a by-product, we establish some Gaussian upper estimates.

A Coupled Unified Difference Method for Numerical Solution of the Initial Boundary Value Fourth Order Parabolic Problems

A Coupled Unified Difference Method for Numerical Solution of the Initial Boundary Value Fourth Order Parabolic Problems

Multiblock Mortar Mixed Approach for Second Order Parabolic Problems

In this paper, the multiblock mortar mixed approximation of second order parabolic partial differential equations is considered. In this method, the simulation domain is decomposed into the non-overlapping subdomains (blocks), and a physically-meaningful boundary condition is set on the mortar interface between the blocks. The governing equations hold locally on each subdomain region. The local problems on blocks are coupled by introducing a special approximation space on the interfaces of neighboring subdomains. Each block is locally covered by an independent grid and the standard mixed finite element method is applied to solve the local problem. The unique solvability of the discrete problem is shown, and optimal order convergence rates are established for the approximate velocity and pressure on the subdomain. Furthermore, an error estimate for the interface pressure in mortar space is presented. The numerical experiments are presented to validate the efficiency of the method.

On the computation of the stabilized coefficients for the 1D spectral VMS method

In this work, we study the computation of the stabilized coefficients for the Variational Multi-Scale method with spectral approximation of the sub-scales, applied to 1D problems. The method is based on an extension of the spectral theorem to operators that have an associated base of eigenfunctions, which are orthonormal in weighted \(L^2\) spaces. We study the discretization of both second order elliptic and parabolic problems with the finite element method. The spectral VMS method is characterized as a standard VMS method with stabilized coefficients issued form the eigenfunctions of the sub-grid problem, that are computed analytically. We derive an off-line/on-line strategy for the computation of the stabilized coefficients. This allows a fast solution of the spectral VMS method, similar to that of the standard VMS one. We display some numerical tests for the stationary and evolutive one-dimensional advection–diffusion equations, in which observe super-convergence effects at the grid nodes.

Local discontinuous Galerkin methods for parabolic interface problems with homogeneous and non-homogeneous jump conditions

In this paper, we present the local discontinuous Galerkin (LDG) methods for solving second order parabolic interface problems in two-dimensional convex polygonal domains with homogeneous and non-homogeneous jump conditions, respectively. We start analyzing from the homogeneous problems. After giving the semi discrete LDG method, we derive its stability and a prior error estimate. Then, after a series of derivation, we obtain the LDG method for the non-homogeneous problems, which takes the same formulation as the homogeneous case with a special choice of numerical fluxes that incorporate the jump conditions across interface. So its analysis can be easily obtained within the framework developed for the homogeneous case. It is an advantage of the LDG method that it provides a natural framework to enforce the discontinuities in both the potential and the flux weakly in the discrete formulation provided that the triangulation of the domain is fitted to the interface. Numerical experiments are conducted to confirm our theoretical analysis with the help of the explicit temporal discretization and show that the LDG methods have the optimal and suboptimal convergence rate in the L2 norms for the potential and the flux, respectively. In addition, they also indicate that the LDG methods possess the same convergence properties in the L∞ norms. What is more, to avoid the strict time-step restriction of explicit schemes, the implicit integration factor (IIF) method is employed, which not only relaxes the time step to Δt=O(hmin) but also remains the important property of LDG method that the computation can proceed element by element and avoid solving a global system of algebraic equations as the standard implicit schemes do. We provide numerical examples to illustrate that the numerical scheme combined LDG method with IIF method indeed reduces the computational cost greatly.

A Discontinuous Finite Volume Element Method Based on Bilinear Trial Functions

We propose a discontinuous finite volume element (DFVE) method for second order elliptic and parabolic problems. Discontinuous bilinear functions are used as the trial functions. We give the stability analysis of this DFVE method and derive the optimal error estimates in the broken [Formula: see text]-norm. Specifically, the optimal [Formula: see text]-error is obtained for the first time for the bilinear DFVE methods solving elliptic and parabolic problems.

Global and Interior Pointwise best Approximation Results for the Gradient of Galerkin Solutions for Parabolic Problems

In this paper we establish best approximation property of fully discrete Galerkin solutions of second order parabolic problems on convex polygonal and polyhedral domains in the $L^\infty(I;W^{1,\infty}(\Om))$ norm. The discretization method consists of continuous Lagrange finite elements in space and discontinuous Galerkin methods of arbitrary order in time. The method of the proof differs from the established fully discrete error estimate techniques and uses only elliptic results and discrete maximal parabolic regularity for discontinuous Galerkin methods established by the authors in \cite{LeykekhmanD_VexlerB_2016b}. In addition, the proof does not require any relationship between spatial mesh sizes and time steps. We also establish interior best approximation property that shows more local dependence of the error at a point.

Linear higher order parabolic problems in locally uniform Lebesgue's spaces

Linear 2m-th order uniformly elliptic operators are shown to generate semigroups of bounded linear operators with suitable smoothing properties in scales of locally uniform Bessel's and Lebesgue's spaces.

Fast Multilevel Solvers for a Class of Discrete Fourth Order Parabolic Problems

In this paper, we study fast iterative solvers for the solution of fourth order parabolic equations discretized by mixed finite element methods. We propose to use consistent mass matrix in the discretization and use lumped mass matrix to construct efficient preconditioners. We provide eigenvalue analysis for the preconditioned system and estimate the convergence rate of the preconditioned GMRes method. Furthermore, we show that these preconditioners only need to be solved inexactly by optimal multigrid algorithms. Our numerical examples indicate that the proposed preconditioners are very efficient and robust with respect to both discretization parameters and diffusion coefficients. We also investigate the performance of multigrid algorithms with either collective smoothers or distributive smoothers when solving the preconditioner systems.

Pointwise Best Approximation Results for Galerkin Finite Element Solutions of Parabolic Problems

In this paper we establish a best approximation property of fully discrete Galerkin finite element solutions of second order parabolic problems on convex polygonal and polyhedral domains in the $L^\infty$ norm. The discretization method uses of continuous Lagrange finite elements in space and discontinuous Galerkin methods in time of an arbitrary order. The method of proof differs from the established fully discrete error estimate techniques and for the first time allows to obtain such results in three space dimensions. It uses elliptic results, discrete resolvent estimates in weighted norms, and the discrete maximal parabolic regularity for discontinuous Galerkin methods established by the authors in [16]. In addition, the proof does not require any relationship between spatial mesh sizes and time steps. We also establish a local best approximation property that shows a more local behavior of the error at a given point.

Blow-up solutions, global existence, and exponential decay estimates for second order parabolic problems

In this paper, we study the blow-up solutions, global existence, and exponential decay estimates for a class of second order parabolic problems with Dirichlet boundary conditions. By constructing auxiliary functions and using maximum principles, the sufficient conditions for the existence of the blow-up solution, the sufficient conditions for the global existence of the solution, an upper bound for the ‘blow-up time’, and some explicit exponential decay bounds for the solution and its derivatives are specified.

Adaptive discontinuous Galerkin approximations to fourth order parabolic problems

An adaptive algorithm, based on residual type a posteriori indicators of errors measured in $L^{\infty}(L^2)$ and $L^2(L^2)$ norms, for a numerical scheme consisting of implicit Euler method in time and discontinuous Galerkin method in space for linear parabolic fourth order problems is presented. The a posteriori analysis is performed for convex domains in two and three space dimensions for local spatial polynomial degrees $r\ge 2$. The a posteriori estimates are then used within an adaptive algorithm, highlighting their relevance in practical computations, which results into substantial reduction of computational effort.

Adaptive discontinuous Galerkin approximations to fourth order parabolic problems

An adaptive algorithm, based on residual type a posteriori indicators of errors measured in $L^{\infty}(L^2)$ and $L^2(L^2)$ norms, for a numerical scheme consisting of implicit Euler method in time and discontinuous Galerkin method in space for linear parabolic fourth order problems is presented. The a posteriori analysis is performed for convex domains in two and three space dimensions for local spatial polynomial degrees $r\ge 2$. The a posteriori estimates are then used within an adaptive algorithm, highlighting their relevance in practical computations, which results into substantial reduction of computational effort.

Critical and supercritical higher order parabolic problems in [formula omitted

Due to the lack of the maximum principle the analysis of higher order parabolic problems in RN is still not as complete as the one of the second-order reaction–diffusion equations. While the critical exponents and then a dissipative mechanism in the subcritical case have already been satisfactorily described (see Cholewa and Rodriguez-Bernal (2012)), for problems in the critical or supercritical regime the questions concerning well or ill-posedness, as well as possible dissipative properties of the solutions, have not yet been satisfactorily answered. This article is devoted to the analysis of the higher order parabolic problems in RN in the latter case. Focusing on the critical and supercritical regimes we give sufficient “good”-sign conditions proving that the problem is then globally well posed in L2(RN) and even possesses a compact global attractor. On the other hand, for supercritically growing “bad”-signed nonlinearities we show that the problem is ill-posed.

An Adaptive Least-Squares Mixed Finite Element Method for Fourth Order Parabolic Problems

A least-squares mixed finite element (LSMFE) method for the numerical solution of fourth order parabolic problems analyzed and developed in this paper. The Ciarlet-Raviart mixed finite element space is used to approximate. The a posteriori error estimator which is needed in the adaptive refinement algorithm is proposed. The local evaluation of the least-squares functional serves as a posteriori error estimator. The posteriori errors are effectively estimated. The convergence of the adaptive least-squares mixed finite element method is proved.

A posteriori error estimates by recovered gradients in parabolic finite element equations

This paper considers a posteriori error estimates by averaged gradients in second order parabolic problems. Fully discrete schemes are treated. The theory from the elliptic case as to when such estimates are asymptotically exact, on an element, is carried over to the error on an element at a given time. The basic principle is that the elliptic theory can be extended to the parabolic problems provided the time-step error is smaller than the space-discretization error. Numerical illustrations confirming the theoretical results are given. Our results are not practical in the sense that various constants can not be estimated realistically. They are conceptual in nature.