Galerkin Methods For Parabolic Problems Research Articles

Adaptive non-hierarchical Galerkin methods for parabolic problems with application to moving mesh and virtual element methods

We present a posteriori error estimates for inconsistent and non-hierarchical Galerkin methods for linear parabolic problems, allowing them to be used in conjunction with very general mesh modification for the first time. We treat schemes which are non-hierarchical in the sense that the spatial Galerkin spaces between time-steps may be completely unrelated from one another. The practical interest of this setting is demonstrated by applying our results to finite element methods on moving meshes and using the estimators to drive an adaptive algorithm based on a virtual element method on a mesh of arbitrary polygons. The a posteriori error estimates, for the error measured in the [Formula: see text] and [Formula: see text] norms, are derived using the elliptic reconstruction technique in an abstract framework designed to precisely encapsulate our notion of inconsistency and non-hierarchicality and requiring no particular compatibility between the computational meshes used on consecutive time-steps, thereby significantly relaxing this basic assumption underlying previous estimates.

An efficient solver for space–time isogeometric Galerkin methods for parabolic problems

In this work we focus on the preconditioning of a Galerkin space–time isogeometric discretization of the heat equation. Exploiting the tensor product structure of the basis functions in the parametric domain, we propose a preconditioner that is the sum of Kronecker products of matrices and that can be efficiently applied thanks to an extension of the classical Fast Diagonalization method. The preconditioner is robust w.r.t. the polynomial degree of the spline space and the time required for the application is almost proportional to the number of degrees-of-freedom, for a serial execution. By incorporating some information on the geometry parametrization and on the equation coefficients, we keep high efficiency with non-trivial domains and variable thermal conductivity and heat capacity coefficients.

$hp$-Version Space-Time Discontinuous Galerkin Methods for Parabolic Problems on Prismatic Meshes

We present a new $hp$-version space-time discontinuous Galerkin (dG) finite element method for the numerical approximation of parabolic evolution equations on general spatial meshes consisting of polygonal/polyhedral (polytopic) elements, giving rise to prismatic space-time elements. A key feature of the proposed method is the use of space-time elemental polynomial bases of total degree, say $p$, defined in the physical coordinate system, as opposed to standard dG time-stepping methods whereby spatial elemental bases are tensorized with temporal basis functions. This approach leads to a fully discrete $hp$-dG scheme using fewer degrees of freedom for each time step, compared to dG time-stepping schemes employing a tensorized space-time basis, with acceptable deterioration of the approximation properties. A second key feature of the new space-time dG method is the incorporation of very general spatial meshes consisting of possibly polygonal/polyhedral elements with an arbitrary number of faces. A priori er...

A splitting mixed space-time discontinuous Galerkin method for parabolic problems

A splitting mixed space-time discontinuous Galerkin method is formulated to solve a class of parabolic problems. This method, in which the stress equation is separated from displacement equation, is based on mixed method and space-time discontinuous finite element method which is discontinuous in time and continuous in space. By a splitting technique, the stress equation is separated from the stress-displacement coupled system. The finite element approximation of the stress is solved by time discontinuous Galerkin method with high accuracy. Then, if required, the discrete displacement function is also solved by the time discontinuous Galerkin method. The convergence of the scheme is analyzed by the technique of combining finite difference and finite element methods. The optimal priori error estimates in norm for displacement and in norm and norm for stress are derived, respectively. Numerical experiments are presented to confirm theoretical results.

A Posteriori Error Control for Discontinuous Galerkin Methods for Parabolic Problems

We derive energy-norm a posteriori error bounds for an Euler time-stepping method combined with various spatial discontinuous Galerkin schemes for linear parabolic problems. For accessibility, we address first the spatially semidiscrete case and then move to the fully discrete scheme by introducing the implicit Euler time-stepping. All results are presented in an abstract setting and then illustrated with particular applications. This enables the error bounds to hold for a variety of discontinuous Galerkin methods, provided that energy-norm a posteriori error bounds for the corresponding elliptic problem are available. To use this method in practice, we apply it to the interior penalty discontinuous Galerkin method, for which new a posteriori error bounds are derived. For the analysis of the time-dependent problems we use the elliptic reconstruction technique, and we deal with the nonconforming part of the error by deriving appropriate computable a posteriori bounds for it. We illustrate the theory with a series of numerical experiments aimed at (1) exploring practically the reliability and efficiency of the derived a posteriori estimates, and (2) testing them in an adaptive algorithm implementation.

A discontinuous Galerkin Method for parabolic problems with modified hp-finite element approximation technique

A recent paper [Hideaki Kaneko, Kim S. Bey, Gene J.W. Hou, Discontinuous Galerkin finite element method for parabolic problems, preprint November 2000, NASA] is generalized to a case where the spatial region is taken in R 3. The region is assumed to be a thin body, such as a panel on the wing or fuselage of an aerospace vehicle. The traditional h- as well as hp-finite element methods are applied to the surface defined in the x– y variables, while, through the thickness, the technique of the p-element is employed. Time and spatial discretization scheme developed in Kaneko et al. (2000), based upon an assumption of certain weak singularity of ∥ u t ∥ 2, is used to derive an optimal a priori error estimate for the current method.

Convergence of a step-doubling Galerkin method for parabolic problems

We analyze a single step method for solving second-order parabolic initial--boundary value problems. The method uses a step-doubling extrapolation scheme in time based on backward Euler and a Galerkin approximation in space. The technique is shown to be a second-order correct approximation in time. Since step-doubling can be used as a mechanism for step-size control, the analysis is done for variable time steps. The stability properties of step-doubling are contrasted with t hose of Crank-Nicolson, and more generally those of extrapolated theta-weighted schemes.

Least-Squares Galerkin Methods for Parabolic Problems II: The Fully Discrete Case and Adaptive Algorithms

In this second part of our series on least-squares Galerkin methods for parabolic initial-boundary value problems we study the full discretization in time and space. These methods are based on the minimization of a least-squares functional for an equivalent first-order system over space and time with respect to suitable discrete spaces. Based on the analysis of the semi-discretization in time carried out in the first part, we construct a posteriori error estimators for the approximation error components associated with time and space. For the space discretization error we use a hierarchical basis estimator with respect to the functional minimization problem. We prove a strengthened Cauchy--Schwarz inequality between coarse and hierarchical surplus space and a bound on the condition number of the auxiliary problem, both uniform in the mesh-size h and the time-step $\tau$ implying efficiency and reliability of the error estimator. This allows for an adaptive strategy for both the time-step size and the spatial triangulation, keeping a proper balance between these two components. Numerical experiments illustrate the performance of the resulting adaptive schemes in time and space.

Least-Squares Galerkin Methods for Parabolic Problems I: Semidiscretization in Time

This is the first part of a series of papers on least-squares Galerkin methods for parabolic initial-boundary value problems. These methods are based on the minimization of a least-squares functional for an equivalent first-order system over space and time with respect to suitable discrete spaces. This paper presents the derivation and analysis of one-step methods for semi-discretization in time from least-squares principles for linear parabolic problems. One of the most important features of the least-squares methodology is the built-in a posteriori estimate for the approximation error. This is a consequence of the equivalence of the least-squares functional to the consistency error associated with a time-step, measured in an appropriate norm. For the presentation in this paper, we focus our attention on the specific combination of piecewise linear, not necessarily continuous, functions with continuous piecewise linear functions for the flux and scalar variable, respectively. For the resulting method, a convergence result is shown for the scalar variable and the associated flux.

On the Application of Extrapolation, Deferred Correction and Defect Correction to Discrete-time Galerkin Methods for Parabolic Problems

On the Application of Extrapolation, Deferred Correction and Defect Correction to Discrete-time Galerkin Methods for Parabolic Problems

Negative norm estimates and superconvergence in Galerkin methods for parabolic problems

Negative norm error estimates for semidiscrete Galerkin-finite element methods for parabolic problems are derived from known such estimates for elliptic problems and applied to prove superconvergence of certain procedures for evaluating point values of the exact solution and its derivatives.

Negative Norm Estimates and Superconvergence in Galerkin Methods for Parabolic Problems

Negative Norm Estimates and Superconvergence in Galerkin Methods for Parabolic Problems

On the extrapolation of Galerkin methods for parabolic problems

The use of Richardson extrapolation in conjunction with several discrete-time Galerkin methods for the approximate solution of parabolic initialboundary value problems is investigated. It is shown that the extrapolation of certain two- and three-level Galerkin approximations which arep th order correct in time yields an improvement ofp orders of accuracy in time per extrapolation, wherep=1, 2. Both linear and quasilinear problems are considered.