Galerkin Approximation In Space Research Articles

A spectral projection based method for the numerical solution of wave equations with memory

In this paper, we compare two approaches to numerically approximate the solution of second-order Gurtin–Pipkin type of integro-differential equations. Both methods are based on a high-order Discontinous Galerkin approximation in space and the numerical inverse Laplace transform. In the first approach, we use functional calculus and the inverse Laplace transform to represent the solution. The spectral projections are then numerically computed and the approximation of the solution of the time-dependent problem is given by a summation of terms that are the product of projections of the data and the inverse Laplace transform of scalar functions. The second approach is the standard inverse Laplace transform technique. We show that the approach based on spectral projections can be very efficient when several time points are computed, and it is particularly interesting for parameter-dependent problems where the data or the kernel depends on a parameter.

Numerical approximation of stochastic evolution equations: Convergence in scale of Hilbert spaces

The present paper is devoted to the numerical approximation of an abstract stochastic nonlinear evolution equation in a separable Hilbert space {$\mathrm{H}$}. Examples of equations which fall into our framework include the GOY and Sabra shell models and { a class of nonlinear heat equations.} The space-time numerical scheme is defined in terms of a Galerkin approximation in space and a { semi-implicit Euler--Maruyama scheme in time}. {We prove the convergence in probability of our scheme by means of an estimate of the error on a localized set of arbitrary large probability.} Our error estimate is shown to hold in a more regular space $\mathrm{V}_{\beta}\subset \mathrm{H}$ with $\beta \in [0,\frac14)$ and { that the explicit rate of convergence of our scheme depends on this parameter $\beta$. }

A novel model order reduction framework via staggered reduced basis space-time finite elements in linear first order transient systems

A novel model order reduction framework for space and time domain discretizations is proposed. Iterative convergence of a Galerkin approximation in space and a Least Squares Petrov Galerkin approximation in time is obtained through a staggered reduced basis method in space-time. In every iteration, one of the two domains (space or time) is refined; and the other is reduced and a posteriori error indicators in space and time are used to drive the convergence iterations. Numerical results for 2D heat transfer and convection-diffusion problems demonstrate the significant computational efficiency of the proposed methodology. Comparisons of wall-clock times and solution accuracy with traditional time integration algorithms has been presented to validate the efficacy of the proposed framework and demonstrate computational savings of an order of magnitude.

Numerical solution of the Burgers equation with Neumann boundary noise

In this paper we investigate the numerical solution of the one-dimensional Burgers equation with Neumann boundary noise. For the discretization scheme we use the Galerkin approximation in space and the exponential Euler method in time. The impact of the boundary noise on the solution is discussed in several numerical examples. Moreover, we analyze and illustrate some properties of the stochastic term and study the convergence numerically.

Space-Time Discontinuous Galerkin Discretizations for Linear First-Order Hyperbolic Evolution Systems

Abstract We introduce a space-time discretization for linear first-order hyperbolic evolution systems using a discontinuous Galerkin approximation in space and a Petrov–Galerkin scheme in time. We show well-posedness and convergence of the discrete system. Then we introduce an adaptive strategy based on goal-oriented dual-weighted error estimation. The full space-time linear system is solved with a parallel multilevel preconditioner. Numerical experiments for the linear transport equation and the Maxwell equation in 2D underline the efficiency of the overall adaptive solution process.

The $L^2$-Projection and Quasi-Optimality of Galerkin Methods for Parabolic Equations

We consider linear parabolic initial-boundary value problems and analyze Galerkin approximation in space. With the help of the inf-sup theory, we derive quasi-optimality results with respect to norms that arise from the standard weak formulation and from a formulation requiring only integrability in time. Moreover, we reveal that the $H^1$-stability of the $L^2$-projection is not only sufficient but also necessary for these results. As application, we consider conforming finite element approximation in space and derive a priori error bounds in terms of the local meshsize and piecewise regularity. The regularity is the minimal one indicated by approximation theory and matches regularity results for linear parabolic problems.

A strategy to suppress recurrence in grid-based Vlasov solvers

In this paper we propose a strategy to suppress the recurrence effect present in grid-based Vlasov solvers. This method is formulated by introducing a cutoff frequency in Fourier space. Since this cutoff only has to be performed after a number of time steps, the scheme can be implemented efficiently and can relatively easily be incorporated into existing Vlasov solvers. Furthermore, the scheme proposed retains the advantage of grid-based methods in that high accuracy can be achieved. This is due to the fact that in contrast to the scheme proposed by Abbasi et al. no statistical noise is introduced into the simulation. We will illustrate the utility of the method proposed by performing a number of numerical simulations, including the plasma echo phenomenon, using a discontinuous Galerkin approximation in space and a Strang splitting based time integration.

Convolution-in-Time Approximations of Time Domain Boundary Integral Equations

We present a new temporal approximation scheme for the boundary integral formulation of time-dependent scattering problems which can be combined with either collocation or Galerkin approximation in space. It uses the backward-in-time framework introduced in [P. J. Davies and D. B. Duncan, Convolution Spline Approximations of Volterra Integral Equations, www.mathstat.strath.ac.uk/research/reports/2012 (2012)] with new temporal basis functions which share some properties with radial basis function multiquadrics. We analyze the stability and convergence properties of the new scheme for associated Volterra integral equations and perform extensive numerical tests for scattering from flat polygonal plates and open and closed cubes and spheres, which demonstrate effectiveness of this approach.

Convergence Analysis of a Discontinuous Galerkin/Strang Splitting Approximation for the Vlasov--Poisson Equations

A rigorous convergence analysis of the Strang splitting algorithm with a discontinuous Galerkin approximation in space for the Vlasov--Poisson equations is provided. It is shown that under suitable assumptions the error is of order $\mathcal{O}(\tau^2+h^q +h^q / \tau)$, where $\tau$ is the size of a time step, $h$ is the cell size, and $q$ the order of the discontinuous Galerkin approximation. In order to investigate the recurrence phenomena for approximations of higher order as well as to compare the algorithm with numerical results already available in the literature a number of numerical simulations are performed.

A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data

This work proposes and analyzes a stochastic collocation method for solving elliptic partial differential equations with random coefficients and forcing terms. These input data are assumed to depend on a finite number of random variables. The method consists of a Galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space, and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach. It treats easily a wide range of situations, such as input data that depend nonlinearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate exponential convergence of the “probability error” with respect to the number of Gauss points in each direction of the probability space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method. Finally, we include a section with developments posterior to the original publication of this work. There we review sparse grid stochastic collocation methods, which are effective collocation strategies for problems that depend on a moderately large number of random variables.

Adaptive local discontinuous Galerkin approximation to Richards’ equation

We propose a spatially and temporally adaptive solution to Richards’ equation based upon a local discontinuous Galerkin approximation in space and a high-order, backward difference method in time. We cast our approach in terms of a general, decoupled adaption algorithm based upon operators. We define non-unique instances of all operators to result in an adaption method from within the general class of methods that is defined. We formally decouple the spatial adaption from the temporal adaption using a method of lines approach and limit the temporal truncation error so that the total error is dominated by the spatial component. We use a multiple grid approach to guide adaption and support the data structures. Spatial adaption decisions are based upon error and regularity indicators, which are economical to compute. The resultant methods are compared for two test problems. The results show that the proposed adaption methods are superior to methods that adapt only in time and that in cases in which the problem has sufficient smoothness, adapting the order of the elements in addition to the grid spacing can further improve the efficiency of this robust solution approach.

A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data

In this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms (input data of the model). The input data are assumed to depend on a finite number of random variables. The method consists in a Galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach. It can be seen as a generalization of the stochastic Galerkin method proposed in [I. Babuška, R. Tempone, and G. E. Zouraris, SIAM J. Numer. Anal., 42 (2004), pp. 800–825] and allows one to treat easily a wider range of situations, such as input data that depend nonlinearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate exponential convergence of the “probability error” with respect to the number of Gauss points in each direction in the probability space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the 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.

A posteriori error estimates for space–time finite element approximation of quasistatic hereditary linear viscoelasticity problems

We give a space–time Galerkin finite element discretisation of the quasistatic compressible linear viscoelasticity problem as described by an elliptic partial differential equation with a fading memory Volterra integral. The numerical scheme consists of a continuous Galerkin approximation in space based on piecewise polynomials of degree p>0 (c G( p)), with a discontinuous Galerkin piecewise constant (d G(0)) or linear (d G(1)) approximation in time. A posteriori Galerkin-error estimates are derived by exploiting the Galerkin framework and optimal stability estimates for a related dual backward problem. The a posteriori error estimates are quite flexible: strong L p -energy norms of the errors are estimated using time derivatives of the residual terms when the data are smooth, while weak-energy norms are used when the data are non-smooth (in time). We also give upper bounds on the d G(0)c G(1) a posteriori error estimates which indicate optimality. However, a complete analysis is not given.

A Variable Time Step Method for an Age-Dependent Population Model with Nonlinear Diffusion

We propose a method for solving a model of age-dependent population diffusion with random dispersal. This method, unlike previous methods, allows for variable time steps and independent age and time discretizations. We use a moving age discretization that transforms the problem to a coupled system of parabolic equations. The system is then solved by backward differences in time and a Galerkin approximation in space; the equations that need to be solved at each step treat each age group separately. A priori L2 error estimates are obtained by an energy analysis. These estimates are superconvergent in the age variable. We present a postprocessing technique which capitalizes on the superconvergence.

Finite element methods of the two nonlinear integro-differential equations

Transformation of dependent variables, such as, the Kirchhoff transformation, is a classical tool for solving nonlinear partial differential equations. In Larsson et al., this approach was used in connection with the finite element method. Zlámal used it in Galerkin approximation in space, the Euler method in time, and applied it to the solution of nonlinear heat equations. Here we give justification of the methods for the two nonlinear integro-heat equations in the case that the discretization is carried through by piecewise linear polynomials in space and by the implicit Euler method in time with quadrature rules which require the storage of relatively few values of the solution. The estimate of the discretization error in the maximum norm is introduced. The results depend on superconvergence in the gradient for a related elliptic problem, which is shown to hold only for very special meshes.

Simulation of continuously deforming parabolic problems by Galerkin finite‐elements method

A general numerical finite element scheme is described for parabolic problems with phase change wherein the elements of the domain are allowed to deform continuously. The scheme is based on the Galerkin approximation in space, and finite difference approximation for the time derivatives. The numerical scheme is applied to the two‐phase Stefan problems associated with the melting and solidification of a substance. Basic functions based on Hermite polynomials are used to allow exact specification of flux‐latent heat balance conditions at the phase boundary. Numerical results obtained by this scheme indicates that the method is stable and produces an accurate solutions for the heat conduction problems with phase change; even when large time steps used. The method is quite general and applicable for a variety of problems involving transition zones and deforming regions, and can be applied for one multidimensional problems.

Optimal error estimates for linear parabolic problems under minimal regularity assumptions

We study the approximation of linear parabolic problems by means of Galerkin approximation in space and θ-method in time. The error is evaluated in norms of typeHtδ (H1x) ⋂Htδ+1/2 (Lx2) for |δ|≤1/2. We prove error estimates which are optimal with respect to the regularity assumptions on the right-hand side of the equation.

Continuously deforming finite elements for the solution of parabolic problems, with and without phase change

AbstractA number of transport problems are complicated by the presence of physically important transition zones where quantities exhibit steep gradients and special numerical care is required. When the location of such a transition zone changes as the solution evolves through time, use of a deforming numerical mesh is appropriate in order to preserve the proper numerical features both within the transition zone and at its boundaries.A general finite element solution method is described wherein the elements are allowed to deform continuously, and the effects of this deformation are accounted for exactly. The method is based on the Galerkin approximation in space, and uses finite difference approximations for the time derivatives. In the absence of element deformation, the method reduces to the conventional Galerkin formulation.The method is applied to the two‐phase Stefan problem associated with the melting and solidification of A substance. The interface between the solid and liquid phase form an internal moving boundary, and latent heat effects are accounted for in the associated boundary condition. By allowing continuous mesh deformation, as dictated by this boundary condition, the moving boundary always lies on element boundaries. This circumvents the difficulties inherent in interpolation of parameters and dependent variables across regions where those quantities change abruptly.Basis functions based on Hermite polynomials are used, to allow exact specification of the flux‐latent heat balance condition at the phase boundary. Analytic solutions for special cases provide tests of the method.