Semidiscrete Galerkin Approximations Research Articles

Error Constants for the Semi-Discrete Galerkin Approximation of the Linear Heat Equation

In this paper, we propose L^2(J;H^1_0(Omega )) and L^2(J;L^2(Omega )) norm error estimates that provide the explicit values of the error constants for the semi-discrete Galerkin approximation of the linear heat equation. The derivation of these error estimates shows the convergence of the approximation to the weak solution of the linear heat equation. Furthermore, explicit values of the error constants for these estimates play an important role in the computer-assisted existential proofs of solutions to semi-linear parabolic partial differential equations. In particular, the constants provided in this paper are better than the existing constants and, in a sense, the best possible.

A NUMERICAL STUDY OF A HOMOGENEOUS BEAM WITH A TIP MASS

In this paper, we prove the existence and uniqueness of the weak solution of a flexible beam that is clamped at one end and free at the other; a mass is also attached to the free end of the beam. Also, we construct a finite element method, based on piecewise cubic Hermitian shape functions. Next, we derive error estimates for the semi-discrete Galerkin approximations. The results are derived from \cite{BS}. Finally, we implement the results of numerical schemes developed.

On the generalized Burgers-Huxley equation: Existence, uniqueness, regularity, global attractors and numerical studies

<p style='text-indent:20px;'>In this work, we consider the forced generalized Burgers-Huxley equation and establish the existence and uniqueness of a global weak solution using a Faedo-Galerkin approximation method. Under smoothness assumptions on the initial data and external forcing, we also obtain further regularity results of weak solutions. Taking external forcing to be zero, a positivity result as well as a bound on the classical solution are also established. Furthermore, we examine the long-term behavior of solutions of the generalized Burgers-Huxley equations. We first establish the existence of absorbing balls in appropriate spaces for the semigroup associated with the solutions and then show the existence of a global attractor for the system. The inviscid limits of the Burgers-Huxley equations to the Burgers as well as Huxley equations are also discussed. Next, we consider the stationary Burgers-Huxley equation and establish the existence and uniqueness of weak solution by using a Faedo-Galerkin approximation technique and compactness arguments. Then, we discuss about the exponential stability of stationary solutions. Concerning numerical studies, we first derive error estimates for the semidiscrete Galerkin approximation. Finally, we present two computational examples to show the convergence numerically.

Analyticity, Regularity, and Generalized Polynomial Chaos Approximation of Stochastic, Parametric Parabolic Two-Scale Partial Differential Equations

We study two-scale parabolic partial differential equations whose coefficient is stochastic and depends linearly on a sequence of pairwise independent random variables which are uniformly distributed in a compact interval. We cast the problem into a deterministic two-scale parabolic problem which depends on a sequence of real parameters in a compact interval. Passing to the limit when the microscale tends to zero, using two-scale homogenization, we obtain the parametric two-scale homogenized problem. This problem contains the solution to the homogenized equation which describes the solution to the original two-scale parabolic problem macroscopically, and the corrector which encodes the microscopic information. The solution of this two-scale homogenized equation is represented as a generalized polynomial chaos (gpc) expansion according to a polynomial basis of the L2 space of the parameters. We use a semidiscrete Galerkin approximation which projects the solution into a space of parametric functions which contain a finite number of pre-chosen gpc modes. Analyticity of the solution of the two-scale homogenized equation with respect to the parameters is established. Under mild assumptions, we show the summability of the coefficients of the solution’s gpc expansion. From this, an explicit error estimate for the semidiscrete Galerkin approximation in terms of the number of the chosen gpc modes is derived when these gpc modes are chosen as the N best ones according to the norms of the gpc coefficients. Regularity of the gpc coefficients and summability of their norms in the regularity spaces are also established. Using the solution of the best N term semidiscrete Galerkin approximation, we derive an approximation for the solution of the original two-scale problem, with an explicit convergence rate in terms of the microscopic scale and the number of gpc modes in the Galerkin approximation.

Galerkin finite element method for time-fractional stochastic diffusion equations

In this paper, Galerkin finite element method for solving the time-fractional stochastic diffusion equations with multiplicative noise is proposed and investigated. The pathwise regularity properties of solutions to the semidiscrete Galerkin approximations are demonstrated and the convergence of optimal rates are derived. And also we construct the fully discrete scheme which is based on the approximations of the Mittag–Leffler function and analyze the error estimates of convergence in $$L_{2}$$ -norm space. Finally, numerical results are conducted to confirm our theoretical findings.

A Priori Feedback Estimates for Multiscale Reaction-Diffusion Systems

ABSTRACTWe study the approximation of a multiscale reaction–diffusion system posed on both macroscopic and microscopic space scales. The coupling between the scales is done through micro–macro flux conditions. Our target system has a typical structure for reaction–diffusion flow problems in media with distributed microstructures (also called, double porosity materials). Besides ensuring basic estimates for the convergence of two-scale semi-discrete Galerkin approximations, we provide a set of a priori feedback estimates and a local feedback error estimator that help in designing a distributed-high-errors strategy to allow for a computationally efficient zooming in and out from microscopic structures. The error control on the feedback estimates relies on two-scale-energy, regularity, and interpolation estimates as well as on a fine bookeeping of the sources responsible with the propagation of the (multiscale) approximation errors. The working technique based on a priori feedback estimates is in principle applicable to a large class of systems of PDEs with dual structure admitting strong solutions.

Polynomial approximations of a class of stochastic multiscale elasticity problems

We consider a class of elasticity equations in ${\mathbb R}^d$ whose elastic moduli depend on $n$ separated microscopic scales, are random and expressed as a linear expansion of a countable sequence of random variables which are independently and identically uniformly distributed in a compact interval. The multiscale Hellinger-Reissner problem that allows for computing the stress directly, and the multiscale mixed problem for nearly incompressible isotropic materials are considered. The stochastic problems are studied via deterministic problems that depend on a countable number of real parameters. We study the multiscale homogenized problems that contain all the macroscopic and microscopic information, whose solutions are written as generalized polynomial chaos (gpc) expansions. We approximate these solutions by semidiscrete Galerkin approximating problems that project into the spaces of functions with only a finite number of $N$ gpc modes. We deduce bounds and summability properties for the solutions' gpc expansion coefficients, which imply explicit rates of convergence in terms of $N$ when the gpc modes used for the Galerkin approximation are chosen to correspond to the best $N$ terms in the gpc expansion. For the mixed problem for nearly incompressible materials, the rate of convergence for the best $N$ term approximation is independent of the Lam\'e constants' ratio. We establish parametric correctors in terms of the semidiscrete Galerkin approximations. For two scale problems, an explicit homogenization rate is deduced. Together with the best $N$ term rate, it provides an explicit convergence rate for the correctors of the parametric multiscale problems. For nearly incompressible materials, we obtain a homogenization rate that is independent of the ratio of the Lam\'e constants, so that the error for the corrector is also independent of this ratio.

Optimal error estimates for semidiscrete Galerkin approximations to equations of motion described by Kelvin–Voigt viscoelastic fluid flow model

In this paper, a finite element Galerkin method is applied to equations of motion arising in the Kelvin–Voigt viscoelastic fluid flow model, when the forcing function is in L∞(L2). Some a priori estimates for the exact solution, which are valid uniformly in time as t↦∞ and even uniformly in the retardation time κ an κ↦0 are derived. It is shown that the semidiscrete method admits a global attractor. Further, with the help of a priori bounds and Sobolev–Stokes projection, optimal error estimates for the velocity in L∞(L2) and L∞(H1)-norms and for the pressure in L∞(L2)-norm are established. Since the constants involved in error estimates have an exponential growth in time, therefore, in the last part of the article, under certain uniqueness condition, the error bounds are established which are valid uniformly in time. Finally, some numerical experiments are conducted which confirm our theoretical findings.

Optimal \(L_{2}\) estimates for semidiscrete Galerkin method applied to parabolic integro-differential equations with nonsmooth data

We propose and analyse an alternate approach to a priori error estimates for the semidiscrete Galerkin approximation to a time-dependent parabolic integro-differential equation with nonsmooth initial data. The method is based on energy arguments combined with repeated use of time integration, but without using parabolic-type duality techniques. An optimal \(L_{2}\)-error estimate is derived for the semidiscrete approximation when the initial data is in \(L_{2}\). A superconvergence result is obtained and then used to prove a maximum norm estimate for parabolic integro-differential equations defined on a two-dimensional bounded domain. doi:10.1017/S1446181114000030

OPTIMAL ESTIMATES FOR THE SEMIDISCRETE GALERKIN METHOD APPLIED TO PARABOLIC INTEGRO-DIFFERENTIAL EQUATIONS WITH NONSMOOTH DATA

AbstractWe propose and analyse an alternate approach to a priori error estimates for the semidiscrete Galerkin approximation to a time-dependent parabolic integro-differential equation with nonsmooth initial data. The method is based on energy arguments combined with repeated use of time integration, but without using parabolic-type duality techniques. An optimal$L^2$-error estimate is derived for the semidiscrete approximation when the initial data is in$L^2$. A superconvergence result is obtained and then used to prove a maximum norm estimate for parabolic integro-differential equations defined on a two-dimensional bounded domain.

An Alternate Approach to Optimal L 2-Error Analysis of Semidiscrete Galerkin Methods for Linear Parabolic Problems with Nonsmooth Initial Data

In this article, we propose and analyze an alternate proof of a priori error estimates for semidiscrete Galerkin approximations to a general second order linear parabolic initial and boundary value problem with rough initial data. Our analysis is based on energy arguments without using parabolic duality. Further, it follows the spirit of the proof technique used for deriving optimal error estimates for finite element approximations to parabolic problems with smooth initial data and hence, it unifies both theories, that is, one for smooth initial data and other for nonsmooth data. Moreover, the proposed technique is also extended to a semidiscrete mixed method for linear parabolic problems. In both cases, optimal L 2-error estimates are derived, when the initial data is in L 2. A superconvergence phenomenon is also observed, which is then used to prove L ∞-estimates for linear parabolic problems defined on two-dimensional spatial domain again with rough initial data.

Semidiscrete formulations for transient transport at small time steps

AbstractSolutions of direct time‐integration schemes for transient advection–diffusion–reaction problems that converge in time to conventional semidiscrete formulations may be polluted at small time steps by spurious spatial oscillations. This degradation is not an artifact of the time‐marching scheme, but rather a property of the solution of the semidiscrete Galerkin approximation itself. An analogy to steady advection–diffusion–reaction problems with a modified reaction coefficient by the Rothe method of discretizing in time prior to spatial discretization provides an upper bound on the time step for the onset of spatial instability. Spatial stabilization removes this pathology, leading to stabilized implicit time‐integration schemes that are free of spurious oscillations at small time steps. Copyright © 2007 John Wiley &amp; Sons, Ltd.

Semidiscrete Galerkin Approximation for a Linear Stochastic Parabolic Partial Differential Equation Driven by an Additive Noise

We study the semidiscrete Galerkin approximation of a stochastic parabolic partial differential equation forced by an additive space-time noise. The discretization in space is done by a piecewise linear finite element method. The space-time noise is approximated by using the generalized L2 projection operator. Optimal strong convergence error estimates in the L2 and \(\dot{H}^{-1}\) norms with respect to the spatial variable are obtained. The proof is based on appropriate nonsmooth data error estimates for the corresponding deterministic parabolic problem. The error estimates are applicable in the multi-dimensional case.

On the numerical approach of the enthalpy method for the Stefan problem

AbstractIn this article an error bound is derived for a piecewise linear finite element approximation of an enthalpy formulation of the Stefan problem; we have analyzed a semidiscrete Galerkin approximation and completely discrete scheme based on the backward Euler method and a linearized scheme is given and its convergence is also proved. A second‐order error estimates are derived for the Crank‐Nicolson Galerkin method. In the second part, a new class of finite difference schemes is proposed. Our approach is to introduce a new variable and transform the given equation into an equivalent system of equations. Then, we prove that the difference scheme is second order convergent. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004

Error estimates for a single phase quasilinear stefan problem with a forcing term

In this paper, we apply finite element Galerkin method to a single-phase quasi-linear Stefan problem with a forcing term. We consider the existence and uniqueness of a semidiscrete approximation and optimal error estimates in L2, L∞, H1 and H2 norms for semidiscrete Galerkin approximations are derived.

The Effect of Spatial Quadrature on the Semidiscrete Finite Element Galerkin Method for a Strongly Damped Wave Equation

The purpose of this article is to study the effect of spatial quadrature on the semidiscrete finite element Galerkin approximations to a linear strongly damped wave equation. Based on a nonstandard energy formulation, optimal order error estimates are derived for all time t > 0. More precisely, for the spatially discrete scheme, optimal order error estimates in L 2 and H 1 norms are proved for nonsmooth initial data. Further, quasi-optimal order error estimate is derived in L ∞ norm for nonsmooth initial data.

Error estimates for semidiscrete Galerkin approximation to a time dependent parabolic integro-differential equation with nonsmooth data

In this paper, an attempt has been made to carry over known results for the finite element Galerkin method for a time dependent parabolic equation with nonsmooth initial data to an integro-differential equation of parabolic type. More precisely, for the homogeneous problem a standard energy technique in conjunction with a duality argument is used to obtain an L2-error estimate of order \(\) for the semidiscrete solution when the given initial function is only in L2. Further, for the nonhomogeneous case with zero initial condition, an error estimate of order \(\) uniformly in time is proved, provided that the nonhomogeneous term is in L∞(L2). The present paper provides a complete answer to an open problem posed on p. 106 of the book Finite Element Methods for Integro-differential Equations by Chen and Shih.

Analysis and semidiscrete galerkin approximation of A class of nonlinear hyperbolic optimal control problems

This paper presents an optimization problem involving a system governed by a nonlinear hyperbolic equation with control acting on the domain, in the boundary conditions, or in the initial conditions. The quadratic cost function is the classical one. We present the Galerkin approximation and we demonstrate existence of the weak and strong condensation points of a set of solutions of the approximate optimization problems. Each of these points is a solution of the initial optimization problem

Semi-discrete Galerkin approximation of the single layer equation by general splines

This paper deals with a semi-discrete version of the Galerkin method for the single-layer equation in a plane, in which the outer integral is approximated by a quadrature rule. A feature of the analysis is that it does not require high precision quadrature or exceptional smoothness of the data. Instead, the assumptions on the quadrature rule are that constant functions are integrated exactly, that the rule is based on sufficiently many quadrature points to resolve the approximation space, and that the Peano constant of the rule is sufficiently small. It is then shown that the semi-discrete Galerkin approximation is well posed. For the trial and test spaces we consider quite general piecewise polynomials on quasi-uniform meshes, ranging from discontinuous piecewise polynomials to smoothest splines. Since it is not assumed that the quadrature rule integrates products of basis functions exactly, one might expect degradation in the rate of convergence. To the contrary, it is shown that the semi-discrete Galerkin approximation will converge at the same rate as the corresponding Galerkin approximation in the $H^0$ and $H^{-1}$ norms.

High accuracy analysis for integrodifferential equations

In this paper we study the higher accuracy methods—the extrapolation and defect correction for the semidiscrete Galerkin approximations to the solutions of the parabolic equations. The global extrapolation and the correction approximation of third order, rather than the pointwise extrapolation results, are derived.