Element Method For Parabolic Problems Research Articles

Space-time CutFEM on overlapping meshes I: simple continuous mesh motion

AbstractWe present a cut finite element method for the heat equation on two overlapping meshes: a stationary background mesh and an overlapping mesh that moves around inside/“on top” of it. Here the overlapping mesh is prescribed by a simple continuous motion, meaning that its location as a function of time is continuous and piecewise linear. For the discrete function space, we use continuous Galerkin in space and discontinuous Galerkin in time, with the addition of a discontinuity on the boundary between the two meshes. The finite element formulation is based on Nitsche’s method and also includes an integral term over the space-time boundary between the two meshes that mimics the standard discontinuous Galerkin time-jump term. The simple continuous mesh motion results in a space-time discretization for which standard analysis methodologies either fail or are unsuitable. We therefore employ what seems to be a relatively uncommon energy analysis framework for finite element methods for parabolic problems that is general and robust enough to be applicable to the current setting. The energy analysis consists of a stability estimate that is slightly stronger than the standard basic one and an a priori error estimate that is of optimal order with respect to both time step and mesh size. We also present numerical results for a problem in one spatial dimension that verify the analytic error convergence orders.

A Space-Time Multiscale Mortar Mixed Finite Element Method for Parabolic Equations

We develop a space-time mortar mixed finite element method for parabolic problems. The domain is decomposed into a union of subdomains discretized with nonmatching spatial grids and asynchronous time steps. The method is based on a space-time variational formulation that couples mixed finite elements in space with discontinuous Galerkin in time. Continuity of flux (mass conservation) across space-time interfaces is imposed via a coarse-scale space-time mortar variable that approximates the primary variable. Uniqueness, existence, and stability as well as a priori error estimates for the spatial and temporal errors are established. A space-time nonoverlapping domain decomposition method is developed that reduces the global problem to a space-time coarse-scale mortar interface problem. Each interface iteration involves solving in parallel space-time subdomain problems. The spectral properties of the interface operator and the convergence of the interface iteration are analyzed. Numerical experiments are provided that illustrate the theoretical results and the flexibility of the method for modeling problems with features that are localized in space and time.

A new error analysis and post-processing technique of the lowest-order Raviart–Thomas mixed finite element method for parabolic problems

A new error analysis and post-processing technique of the lowest-order Raviart–Thomas mixed finite element method for parabolic problems

Efficient implementation of mixed finite element methods for parabolic problems

It is well-known that conventional mixed finite element method (FEM) results in a large saddle point system or non-symmetric system. A splitting mixed FEM is introduced by Yang (Numer. Methods Partial Differential Equations 17: 229–249, 2001), which computes the flux variable by solving a symmetric positive definite system. Then, the original unknown is solved based on the computed flux. In this paper, we prove that the splitting mixed FEM proposed by Yang is equivalent to the conventional mixed FEM provided that the initial data is carefully taken. Moreover, we show that the equivalence also holds for more complicated models, such as splitting mixed FEM for the compressible flow in porous media. Therefore, all available theoretical results of conventional mixed FEMs hold for the splitting ones. Numerical comparisons are provided for the conventional mixed FEM and the splitting mixed FEM. Efficiency of the splitting mixed method is clearly demonstrated. Corresponding results can be extended to many other mixed FEMs, such as Brezzi–Douglas–Marini mixed FEM.

A priori and a posteriori error estimates of the weak Galerkin finite element method for parabolic problems

We derive the priori and a posteriori error estimates of the weak Galerkin finite element method with the Crank-Nicolson time discretization for the parabolic equation in this paper. The priori error estimates are deduced based on existing priori error results of the corresponding elliptic projection problem. For the a posteriori error estimates, the elliptic reconstruction technique is introduced to decompose the true error into elliptic error and parabolic error. Then the elliptic part is bounded by the a posteriori error estimates of the auxiliary elliptic reconstruction problem. The a posteriori error estimator is further used to develop the temporal and spatial adaptive algorithm. Numerical results in the uniform and adaptive meshes are provided to validate the proposed estimators.

Stability of Galerkin discretizations of a mixed space–time variational formulation of parabolic evolution equations

Abstract We analyze Galerkin discretizations of a new well-posed mixed space–time variational formulation of parabolic partial differential equations. For suitable pairs of finite element trial spaces, the resulting Galerkin operators are shown to be uniformly stable. The method is compared to two related space–time discretization methods introduced by Andreev (2013, Stability of sparse space-time finite element discretizations of linear parabolic evolution equations. IMA J. Numer. Anal., 33, 242–260) and by Steinbach (2015, Space-time finite element methods for parabolic problems. Comput. Methods Appl. Math., 15, 551–566).

Energy-corrected FEM and explicit time-stepping for parabolic problems

The presence of corners in the computational domain, in general, reduces the regularity of solutions of parabolic problems and diminishes the convergence properties of the finite element approximation introducing a so-called “pollution effect”. Standard remedies based on mesh refinement around the singular corner result in very restrictive stability requirements on the time-step size when explicit time integration is applied. In this article, we introduce and analyse the energy-corrected finite element method for parabolic problems, which works on quasi-uniform meshes, and, based on it, create fast explicit time discretisation. We illustrate these results with extensive numerical investigations not only confirming the theoretical results but also showing the flexibility of the method, which can be applied in the presence of multiple singular corners and a three-dimensional setting. We also propose a fast explicit time-stepping scheme based on a piecewise cubic energy-corrected discretisation in space completed with mass-lumping techniques and numerically verify its efficiency.

Interior Estimates of Semidiscrete Finite Element Methods for Parabolic Problems with Distributional Data

Interior Estimates of Semidiscrete Finite Element Methods for Parabolic Problems with Distributional Data

New a priori analysis of first‐order system least‐squares finite element methods for parabolic problems

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.

Performance of space-time coupled least-squares spectral element methods for parabolic problems

We study the performance of space-time coupled least-squares spectral element method (LSSEM) for parabolic initial boundary value problem (IBVP) for different values of element size h, the time step k, the degree q in the time variable and the degree p in each of the space variables. We divide the space domain into a number of shape regular quadrilaterals of size h and the time step k is proportional to h2. Each shape regular quadrilateral will be mapped to the square (−1,1) × (−1,1). In each square the solution will be defined as a polynomial of degree p for the space variables and degree q for the time variable. For the p version of the method h remains fixed and p increases, for the the h-version of the method p remains fixed and h decreases but for the hp-version of the method h decreases and at the same time p also increases. In this method, k is proportional to h2 (say k = ch2) and for the p-version of the method, q is proportional to p2(say q = c′p2). The performance of the method is discussed for different values of c and c′. The method we discussed here is spectral in both space and time and are non-conforming.

A convergent time–space adaptive dG(s) finite element method for parabolic problems motivated by equal error distribution

A convergent time–space adaptive dG(s) finite element method for parabolic problems motivated by equal error distribution

Parallel split least‐squares mixed finite element method for parabolic problem

Based on overlapping domain decomposition, a new class of parallel split least‐squares (PSLS) mixed finite element methods is presented for solving parabolic problem. The algorithm is fully parallel. In the overlapping domains, the partition of unity is applied to distribute the corrections reasonably, which makes that the new method only needs one or two iteration steps to reach given accuracy at each time step while the classical Schwarz alternating methods need many iteration steps. The dependence of the convergence rate on the spacial mesh size, time increment, iteration times, and subdomains overlapping degree is analyzed. Some numerical results are reported to confirm the theoretical analysis.

Two-scale composite finite element method for parabolic problems with smooth and nonsmooth initial data

We consider a new class of finite elements, called composite finite elements, for the discretization of parabolic problems in a two-dimensional convex polygonal domain. More precisely, an effort has been made in this paper to extend two-scale composite finite element method for elliptic problems to parabolic problems. We analyze both semidiscrete and fully discrete composite finite element methods and derive convergence properties in the $$L^\infty (L^2)$$ and $$L^\infty (H^1)$$ norms for smooth data. Moreover, nonsmooth data error estimates in the $$L^\infty (L^2)$$ norm for positive time is established. Compared to the classical finite element method, the composite finite element method can be viewed as a coarse scale generalization which yields a minimal dimension of the approximation space and the asymptotic order of convergence is preserved on coarse scale meshes which do not resolve the boundary. Numerical results are presented to illustrate the theoretical rates of convergence.

High-order evolving surface finite element method for parabolic problems on evolving surfaces

High-order spatial discretisations and full discretisations of parabolic partial differential equations on evolving surfaces are studied. We prove convergence of the high-order evolving surface finite element method, by showing high-order versions of geometric approximation errors and perturbation error estimates and by the careful error analysis of a modified Ritz map. Furthermore, convergence of full discretisations using backward difference formulae and implicit Runge-Kutta methods are also shown.

A new approach to constructing vector splitting schemes in mixed finite element method for parabolic problems

Abstract A general setting for a new approach to constructing vector splitting schemes in mixed FEM for heat transfer problem is considered. For space approximation Raviart–Thomas finite elements of lowest order are implemented on rectangular (parallelepiped) mesh. The main question discussed is how to implement time discretization so as to obtain efficient numerical algorithms. The key idea of the proposed approach is to use scalar splitting schemes for heat flux divergence. This allows one to carry out accuracy and stability analysis on the basis of the well-known results for underlying scalar splitting schemes. A priori estimates are obtained for vector splitting schemes in two- and three-dimensional cases using the idea of flux decomposition onto divergence-free and potential components at discrete level. It is shown that splitting schemes which have been proposed by other approaches based on Uzawa algorithm’s modification and block triangular factorization are included as special cases by an appropriate choice of the underlying splitting schemes for heat flux divergence. Generalization of the discussed approach is presented for second-order hyperbolic problems. Moreover, the general idea of using splitting schemes for the flux divergence to construct splitting schemes for fluxes can be easily extended to first-order hyperbolic systems, but by now several questions still remain unsolved.

Virtual element methods for parabolic problems on polygonal meshes

The virtual element method (VEM) is a recent technology that can make use of very general polygonal/polyhedral meshes without the need to integrate complex nonpolynomial functions on the elements and preserving an optimal order of convergence. In this article, we develop for the first time, the VEM for parabolic problems on polygonal meshes, considering time‐dependent diffusion as our model problem. After presenting the scheme, we develop a theoretical analysis and show the practical behavior of the proposed method through a large array of numerical tests. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 2110–2134, 2015

Convergence of H 1-Galerkin mixed finite element method for parabolic problems with reduced regularity on initial data

We study the convergence of H1-Galerkin mixed finite element method for parabolic problems in one space dimension. Both semi-discrete and fully discrete schemes are analyzed assuming less regularity on initial data. More precisely, for the spatially discrete scheme, error estimates of order \(\mathcal{O}\)(h2t−1/2) for positive time are established assuming the initial function p0 ∈ H2(Ω) ∩ H01 (Ω). Further, we use energy technique together with parabolic duality argument to derive error estimates of order \(\mathcal{O}\)(h2t−1) when p0 is only in H01 (Ω). A discrete-in-time backward Euler method is analyzed and almost optimal order error bounds are established.

A Fully Discrete Stabilized Mixed Finite Element Method for Parabolic Problems

In this article, we present a new stabilized mixed finite element method based on the less regularity of flux (velocity) for the two-dimensional parabolic problems in practice. The method combines the Crank-Nicolson scheme with a stabilized mixed finite element method which is based on the velocity projection stabilization method by using the lowest equal-order pair for the velocity and pressure. It is shown that the proposed fully discrete stabilized finite element method results in the optimal error estimates in L 2- and H 1-norm for the pressure and suboptimal error estimate in L 2-norm for the velocity. Finally, we give some numerical experiments to verify the efficiency and theoretical results of this method.

A posteriori error estimates for H 1-Galerkin mixed finite-element method for parabolic problems

The purpose of this article is to derive a posteriori error estimates for the H 1-Galerkin mixed finite element method for parabolic problems. We study both semidiscrete and fully discrete a posteriori error analyses using standard energy argument. A fully discrete a posteriori error analysis based on the backward Euler method is analysed and upper bounds for the errors are derived. The estimators yield upper bounds for the errors which are global in space and time. Our analysis is based on residual approach and the estimators are free from edge residuals.

And Norms Error Estimates in Finite Element Method for Linear Parabolic Interface Problems

We derive new a priori error estimates for linear parabolic equations with discontinuous coefficients. Due to low global regularity of the solutions the error analysis of the standard finite element method for parabolic problems is difficult to adopt for parabolic interface problems. A finite element procedure is, therefore, proposed and analyzed in this paper. We are able to show that the standard energy technique of finite element method for non-interface parabolic problems can be extended to parabolic interface problems if we allow interface triangles to be curved triangles. Optimal pointwise-in-time error estimates in the L 2(Ω) and H 1(Ω) norms are shown to hold for the semidiscrete scheme. A fully discrete scheme based on backward Euler method is analyzed and pointwise-in-time error estimates are derived. The interfaces are assumed to be arbitrary shape but smooth for our purpose.