Element Method For Parabolic Equations Research Articles

Optimal convergence analysis of weak Galerkin finite element methods for parabolic equations with lower regularity

Optimal convergence analysis of weak Galerkin finite element methods for parabolic equations with lower regularity

A modified weak Galerkin finite element method for parabolic equations on anisotropic meshes

In this paper, we study a modified weak Galerkin finite element method (MWG-FEM) on anisotropic triangular meshes for a class of parabolic equations. Different from conventional weak Galerkin methods, the MWG-FEM replaces the boundary functions by the average of interior functions, which possesses flexibility in the approximation functions and mesh generation. Moreover, MWG-FEM is compatible with anisotropic meshes and suitable for solving problems with anisotropic property. By using the anisotropic interpolation and projection operators, the optimal order error estimate in L2 norm is derived. Numerical experiments are performed to demonstrate the stability and efficiency of the method.

Maximal regularity of multistep fully discrete finite element methods for parabolic equations

AbstractThis article extends the semidiscrete maximal $L^p$-regularity results in Li (2019, Analyticity, maximal regularity and maximum-norm stability of semi-discrete finite element solutions of parabolic equations in nonconvex polyhedra. Math. Comp., 88, 1--44) to multistep fully discrete finite element methods for parabolic equations with more general diffusion coefficients in $W^{1,d+\beta }$, where $d$ is the dimension of space and $\beta>0$. The maximal angles of $R$-boundedness are characterized for the analytic semigroup $e^{zA_h}$ and the resolvent operator $z(z-A_h)^{-1}$, respectively, associated to an elliptic finite element operator $A_h$. Maximal $L^p$-regularity, an optimal $\ell ^p(L^q)$ error estimate and an $\ell ^p(W^{1,q})$ estimate are established for fully discrete finite element methods with multistep backward differentiation formulae.

A stabilizer free weak Galerkin finite element method for parabolic equation

In this article, a stabilizer free weak Galerkin (SFWG) finite element method for a parabolic partial differential equation is proposed. The goal of using the SFWG method is to make the numerical implementation easier and more efficient. The optimal rate of convergence is derived in both H1 and L2 norms. Numerical experiments are performed to verify the theoretical results.

Error estimates in weak Galerkin finite element methods for parabolic equations under low regularity assumptions

In this paper, we consider the weak Galerkin finite element approximations of second order linear parabolic problems in two dimensional convex polygonal domains under the low regularities of the solutions. Optimal order error estimates in L2(L2) and L2(H1) norms are shown to hold for both the spatially discrete continuous time and the discrete time weak Galerkin finite element schemes, which allow using the discontinuous piecewise polynomials on finite element partitions with the arbitrary shape of polygons with certain shape regularity. The fully discrete scheme is based on first order in time Euler method. We have derived O(hr+1) in L2(L2) norm and O(hr) in L2(H1) norm when the exact solution u∈L2(0,T;Hr+1(Ω))∩H1(0,T;Hr−1(Ω)), for some r≥1. Numerical experiments are reported for several test cases to justify our theoretical convergence results.

A combined finite element method for parabolic equations posted in domains with rough boundaries

In this paper, we consider the solution of parabolic equations posted in domains with rough boundaries using the combined finite element procedure proposed in Xu et al. [A combined finite element method for elliptic problems posted in domains with rough boundaries, J. Comput. Appl. Math. 336 (2018), pp. 235–248] in space and backward-Euler scheme in time. We first prove error estimates for elliptic projection operator for both and by the dual argument which depend on the elliptic auxiliary problem, and then energy error estimates of semi-discrete and fully discrete schemes which have convergence rates about and , where s>0, respectively. Numerical results are provided for parabolic equations in domains with non-oscillating or oscillating boundaries to verify the theoretical findings.

A Constraint Energy Minimizing Generalized Multiscale Finite Element Method for Parabolic Equations

In this paper, we present a constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) for parabolic equations with multiscale coefficients, arising from applications in porous media. We will present the construction of CEM-GMsFEM and rigorously analyze its convergence for the parabolic equations. The convergence rate is characterized by the coarse grid size and the eigenvalue decay of local spectral problems, but is independent of the scale length and contrast of the media. The analysis shows that the method has a first order convergence rate with respect to coarse grid size in the energy norm and second order convergence rate with respect to coarse grid size in $L^2$ norm under some appropriate assumptions. For the temporal discretization, finite difference techniques are used and the convergence analysis of full discrete scheme is given. Moreover, a posteriori error estimator is derived and analyzed. A few numerical results for porous media applications are presented to confirm the theoretical findings and demonstrate the performance of the approach.

Weak Galerkin mixed finite element methods for parabolic equations with memory

We develop a semidiscrete and a backward Euler fully discrete weak Galerkin mixed finite element method for a parabolic differential equation with memory. The optimal order error estimates in both \begin{document}$ |\|·|\| $\end{document} and \begin{document}$ L^2 $\end{document} norms are established based on a generalized elliptic projection. In the numerical experiments, the equation is solved by the weak Galerkin schemes with spaces \begin{document}$ \{[P_{k}(T)]^2, P_{k}(e), P_{k+1}(T)\} $\end{document} for \begin{document}$ k = 0 $\end{document} and the numerical convergence rates confirm the theoretical results.

Discrete maximal regularity and the finite element method for parabolic equations

Maximal regularity is a fundamental concept in the theory of partial differential equations. In this paper, we establish a fully discrete version of maximal regularity for a parabolic equation. We derive various stability results in $L^p(0,T;L^q(\Omega))$ norm, $p,q\in (1,\infty)$ for the finite element approximation with the mass-lumping to the linear heat equation. Our method of analysis is an operator theoretical one using pure imaginary powers of operators and might be a discrete version of G.~Dore and A.~Venni (On the closedness of the sum of two closed operators. \emph{Math.\ Z.}, 196(2):189--201, 1987). As an application, optimal order error estimates in that norm are proved. Furthermore, we study the finite element approximation for semilinear heat equations with locally Lipschitz continuous nonlinearity and offer a new method for deriving optimal order error estimates. Some interesting auxiliary results including discrete Gagliardo-Nirenberg and Sobolev inequalities are also presented.

Adaptive finite element method for parabolic equations with Dirac measure

In this paper we study the adaptive finite element method for parabolic equations with Dirac measure. Two kinds of problems with separate measure data in time and measure data in space are considered. It is well known that the solutions of such kind of problems may exhibit lower regularity due to the existence of the Dirac measure, and thus fit to adaptive FEM for space discretization and variable time steps for time discretization. For both cases we use piecewise linear and continuous finite elements for the space discretization and backward Euler scheme, or equivalently piecewise constant discontinuous Galerkin method, for the time discretization, the a posteriori error estimates based on energy and L2 norms for the fully discrete problems are then derived to guide the adaptive procedure. Numerical results are provided at the end of the paper to support our theoretical findings.

The splitting mixed element method for parabolic equation and its application in chemotaxis model

In this article, we first revisit the splitting positive definite mixed element method for reaction-diffusion equation, in which the mixed system is symmetric positive definite. And then we apply this technique to the variable coefficient parabolic equation and give the corresponding fully-discrete scheme with second-order central difference formula in time. We study the convergence of the semi-discrete and fully-discrete scheme and derive the error estimates. Finally, we extend this method to chemotaxis model and give the corresponding numerical results, which suggests that it has the ability to recover blowing-up solutions.

Sensitivity Analysis of Temperature Field in Domain of Skin Tissue with Respect to Perturbations of Protective Clothing Parameters

The aim of considerations is the numerical modeling of thermal processes proceeding in the system external heat source - protective clothing - air gap - human skin and next the application of sensitivity analysis methods to study the impact of clothing parameters on the transient temperature field in domain of skin tissue. From the mathematical point of view the problem is described by the system of Fourier and Pennes equations determining the transient temperature field in the successive sub-domains. This system is supplemented by the appropriate boundary and initial conditions. Taking into account the geometrical properties of the domain considered, the 1D solution seems to be sufficiently exact (e.g. chest or back). The sensitivity model is constructed on the basis of the so-called direct approach. The equations creating the mathematical model of the process considered are differentiated with respect to selected parameter. At the stage of numerical modeling the 1st scheme of the boundary element method for parabolic equations has been used.

Maximum-norm stability and maximal $$L^{p}$$ L p regularity of FEMs for parabolic equations with Lipschitz continuous coefficients

In this paper, we study the semi-discrete Galerkin finite element method for parabolic equations with Lipschitz continuous coefficients. We prove the maximum-norm stability of the semigroup generated by the corresponding elliptic finite element operator, and prove the space-time stability of the parabolic projection onto the finite element space in $$L^\infty ( Q_T)$$L?(QT) and $$L^p((0,T);L^q(\Omega ))$$Lp((0,T)?Lq(Ω)), $$1

Domain decomposition procedures combined with [formula omitted]-Galerkin mixed finite element method for parabolic equation

Non-overlapping domain decomposition procedures are considered for parabolic equation. These procedures are combined with using H1-Galerkin mixed finite element method in the sub-domains to approximate the primary variable u and its flux σ simultaneously. Explicit calculations are built by using integral mean methods to present the inter-domain boundary conditions for the flux. Thus, the parallelism can be achieved by these procedures. Two approximation schemes are established. Time step constraints are proved necessary to preserve stability, which are less severe than that of fully explicit Galerkin finite element method. The mixed finite element spaces are allowed to be of different polynomial degrees and not subject to the LBB-consistency condition. New nonstandard elliptic projections are defined and analyzed. Optimal error estimates for the variable u in H1-norm and its flux σ in L2-norm and are derived for these schemes. Numerical experiments are presented to confirm the theoretical results.

Eulerian Finite Element Methods for Parabolic Equations on Moving Surfaces

Three new Eulerian finite element methods for parabolic PDEs on a moving surface $\Gamma(t)$ are presented and compared in numerical experiments. These are space-time Galerkin methods, which are derived from a weak formulation in space and time. The trial- and test-spaces contain the traces on the space-time manifold of an outer prismatic finite element space. The numerical experiments show that two of the methods converge with second order with respect to both the time step size and the spatial mesh width.

Convergence of linearized backward Euler–Galerkin finite element methods for the time-dependent Ginzburg–Landau equations with temporal gauge

We present error analysis of fully discrete Galerkin finite element methods for the time-dependent Ginzburg–Landau equations with the temporal gauge, where a linearized backward Euler scheme is used for the time discretization. We prove that the convergence rate is O(τ+hr) if the finite element space of piecewise polynomials of degree r is used. Due to the degeneracy of the problem, the convergence rate is one order lower than the optimal convergence rate of finite element methods for parabolic equations. Numerical examples are provided to support our theoretical analysis.

Weak Galerkin finite element methods for Parabolic equations

AbstractA newly developed weak Galerkin method is proposed to solve parabolic equations. This method allows the usage of totally discontinuous functions in approximation space and preserves the energy conservation law. Both continuous and discontinuous time weak Galerkin finite element schemes are developed and analyzed. Optimal‐order error estimates in both H1 and L2 norms are established. Numerical tests are performed and reported. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

Immersed finite element methods for parabolic equations with moving interface

AbstractThis article presents three Crank‐Nicolson‐type immersed finite element (IFE) methods for solving parabolic equations whose diffusion coefficient is discontinuous across a time dependent interface. These methods can use a fixed mesh because IFEs can handle interface jump conditions without requiring the mesh to be aligned with the interface. These methods will be compared analytically in the sense of accuracy and computational cost. Numerical examples are provided to demonstrate features of these three IFE methods. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

Asymptotic behavior of the finite difference and the finite element methods for parabolic equations

The asymptotic convergence of the solution of the parabolic equation is proved. By the eigenvalues estimation, we obtain that the approximate solutions by the finite difference method and the finite element method are asymptotically convergent. Both methods are considered in continuous time.

Error estimates for a finite volume element method for parabolic equations in convex polygonal domains

AbstractWe analyze the spatially semidiscrete piecewise linear finite volume element method for parabolic equations in a convex polygonal domain in the plane. Our approach is based on the properties of the standard finite element Ritz projection and also of the elliptic projection defined by the bilinear form associated with the variational formulation of the finite volume element method. Because the domain is polygonal, special attention has to be paid to the limited regularity of the exact solution. We give sufficient conditions in terms of data that yield optimal order error estimates in L2 and H1. The convergence rate in the L∞ norm is suboptimal, the same as in the corresponding finite element method, and almost optimal away from the corners. We also briefly consider the lumped mass modification and the backward Euler fully discrete method. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004