Showing papers in "SIAM Journal on Numerical Analysis in 1998"
TL;DR: It is proven that for scalar equations, the LDG methods are L2-stable in the nonlinear case and in the linear case, it is shown that if polynomials of degree k are used, the methods are kth order accurate for general triangulations.
Abstract: In this paper, we study the local discontinuous Galerkin (LDG) methods for nonlinear, time-dependent convection-diffusion systems. These methods are an extension of the Runge--Kutta discontinuous Galerkin (RKDG) methods for purely hyperbolic systems to convection-diffusion systems and share with those methods their high parallelizability, high-order formal accuracy, and easy handling of complicated geometries for convection-dominated problems. It is proven that for scalar equations, the LDG methods are L2-stable in the nonlinear case. Moreover, in the linear case, it is shown that if polynomials of degree k are used, the methods are kth order accurate for general triangulations; although this order of convergence is suboptimal, it is sharp for the LDG methods. Preliminary numerical examples displaying the performance of the method are shown.
2,265 citations
TL;DR: A rigorous proof of the asymptotic optimality of the multigrid method can be given, which shows that convergence does not deteriorate on very fine grids.
Abstract: In this paper we are concerned with the efficient solution of discrete variational problems related to the bilinear form ({\bf curl}\! $\cdot$, {\bf curl}\! $\cdot)_{{\fontsize{6}{0pt}\selectfont\textbf{\textit{L}}}^2(\Omega)}$ + ($\cdot,\cdot)_{{\fontsize{6}{0pt}\selectfont\textbf{\textit{L}}}^2(\Omega)}$ defined on \textbf{\textit{H}}$_0$({\bf curl}; $\Omega$). This is a core task in the time-domain simulation of electromagnetic fields, if implicit timestepping is employed. We rely on Nedelec's \textbf{\textit{H}}({\bf curl}; $\Omega$)-conforming finite elements (edge elements) to discretize the problem.
We construct a multigrid method for the fast iterative solution of the resulting linear system of equations. Since proper ellipticity of the bilinear form is confined to the complement of the kernel of the {\bf curl} operator, Helmholtz decompositions are the key to the design of the algorithm: ${\cal N}$({\bf curl}) and its complement ${\cal N}$({\bf curl})$^{\bot}$ require separate treatment. Both can be tackled by nodal multilevel decompositions, where for the former the splitting is set in the space of discrete scalar potentials.
Under certain assumptions on the computational domain and the material functions, a rigorous proof of the asymptotic optimality of the multigrid method can be given, which shows that convergence does not deteriorate on very fine grids. The results of numerical experiments confirm the practical efficiency of the method.
421 citations
TL;DR: In this paper, Despres et al. introduced a new technique to solve elliptic linear PDEs, called ultra weak variational formulation (UWVF), which is applied to a model wave problem, the two-dimensional Helmholtz problem.
Abstract: A new technique to solve elliptic linear PDEs, called ultra weak variational formulation (UWVF) in this paper, is introduced in [B. Despres, C. R. Acad. Sci. Paris, 318 (1994), pp. 939--944]. This paper is devoted to an evaluation of the potentialities of this technique. It is applied to a model wave problem, the two-dimensional Helmholtz problem. The new method is presented in three parts following the same style of presentation as the classical one of the finite elements method, even though they are definitely conceptually different methods. The first part is committed to the variational formulation and to the continuous problem. The second part defines the discretization process using a Galerkin procedure. The third part actually studies the efficiency of the technique from the order of convergence point of view. This is achieved using theoretical proofs and a series of numerical experiments. In particular, it is proven and shown the order of convergence is lower bounded by a linear function of the number of degrees of freedom. An application to scattering problems is presented in a fourth part.
408 citations
TL;DR: One-dimensional scalar conservation laws with nondecreasing initial conditions and general fluxes were shown to be the appropriate equations to describe large systems of free particles on the real line, which stick under collision with conservation of mass and momentum as mentioned in this paper.
Abstract: One-dimensional scalar conservation laws with nondecreasing initial conditions and general fluxes are shown to be the appropriate equations to describe large systems of free particles on the real line, which stick under collision with conservation of mass and momentum.
346 citations
TL;DR: An adaptive mesh refinement algorithm developed for the Euler equations of gas dynamics has been extended to employ high-resolution wave-propagation algorithms in a more general framework, including hyperbolic equations not in conservation form, problems with source terms or capacity functions, and logically rectangular curvilinear grids.
Abstract: An adaptive mesh refinement algorithm developed for the Euler equations of gas dynamics has been extended to employ high-resolution wave-propagation algorithms in a more general framework. This allows its use on a variety of new problems, including hyperbolic equations not in conservation form, problems with source terms or capacity functions, and logically rectangular curvilinear grids. This framework requires a modified approach to maintaining consistency and conservation at grid interfaces, which is described in detail. The algorithm is implemented in the AMRCLAW package, which is freely available.
296 citations
TL;DR: This paper proves convergence for a generic form of Pseudo-transient continuation and illustrates it with two practical strategies.
Abstract: Pseudo-transient continuation ($\Psi$tc) is a well-known and physically motivated technique for computation of steady state solutions of time-dependent partial differential equations. Standard globalization strategies such as line search or trust region methods often stagnate at local minima. \ptc succeeds in many of these cases by taking advantage of the underlying PDE structure of the problem. Though widely employed, the convergence of \ptc is rarely discussed. In this paper we prove convergence for a generic form of \ptc and illustrate it with two practical strategies.
286 citations
TL;DR: The idea in this approach is to precondition the differential equation before applying the immersed interface method, and to take advantage of fast Poisson solvers on a rectangular region, an intermediate unknown function, the jump in the normal derivative across the interface, is introduced.
Abstract: A fast, second-order accurate iterative method is proposed for the elliptic equation \[ \grad\cdot(\beta(x,y) \grad u) =f(x,y) \] in a rectangular region $\Omega$ in two-space dimensions. We assume that there is an irregular interface across which the coefficient $\beta$, the solution u and its derivatives, and/or the source term f may have jumps. We are especially interested in the cases where the coefficients $\beta$ are piecewise constant and the jump in $\beta$ is large. The interface may or may not align with an underlying Cartesian grid. The idea in our approach is to precondition the differential equation before applying the immersed interface method proposed by LeVeque and Li [ SIAM J. Numer. Anal., 4 (1994), pp. 1019--1044]. In order to take advantage of fast Poisson solvers on a rectangular region, an intermediate unknown function, the jump in the normal derivative across the interface, is introduced. Our discretization is equivalent to using a second-order difference scheme for a corresponding Poisson equation in the region, and a second-order discretization for a Neumann-like interface condition. Thus second-order accuracy is guaranteed. A GMRES iteration is employed to solve the Schur complement system derived from the discretization. A new weighted least squares method is also proposed to approximate interface quantities from a grid function. Numerical experiments are provided and analyzed. The number of iterations in solving the Schur complement system appears to be independent of both the jump in the coefficient and the mesh size.
279 citations
TL;DR: In this paper, the electrostatic interpretation of the Jacobi-Gauss quadrature points is exploited to obtain interpolation points suitable for approximation of smooth functions defined on a simplex.
Abstract: The electrostatic interpretation of the Jacobi--Gauss quadrature points is exploited to obtain interpolation points suitable for approximation of smooth functions defined on a simplex. Moreover, several new estimates, based on extensive numerical studies, for approximation along the line using Jacobi--Gauss--Lobatto quadrature points as the nodal sets are presented.
The electrostatic analogy is extended to the two-dimensional case, with the emphasis being on nodal sets inside a triangle for which two very good matrices of nodal sets are presented. The matrices are evaluated by computing the Lebesgue constants and they share the property that the nodes along the edges of the simplex are the Gauss--Lobatto quadrature points of the Chebyshev and Legendre polynomials, respectively. This makes the resulting nodal sets particularly well suited for integration with conventional spectral methods and supplies a new nodal basis for h-p finite element methods.
257 citations
TL;DR: In this paper, the authors introduce implicitness in stochastic terms of numerical methods for solving stiff Stochastic differential equations and especially a class of fully implicit methods, the balanced methods.
Abstract: This paper introduces some implicitness in stochastic terms of numerical methods for solving stiff stochastic differential equations and especially a class of fully implicit methods, the balanced methods. Their order of strong convergence is proved. Numerical experiments compare the stability properties of these schemes with explicit ones.
226 citations
TL;DR: A local regularization operator on triangular or quadrilateral finite elements built on structured or unstructured meshes is developed and it is proved that it has the same optimal approximation properties as the standard interpolation operator.
Abstract: This paper develops a local regularization operator on triangular or quadrilateral finite elements built on structured or unstructured meshes. This operator is a variant of the regularization operator of Clement; however, ours is constructed via a local projection in a reference domain. We prove in this paper that it has the same optimal approximation properties as the standard interpolation operator, and we present some applications.
199 citations
TL;DR: A two-level method for resolving the nonlinearity in finite element approximations of the equilibrium Navier--Stokes equations produces an approximate solution with the optimal, asymptotic in h, accuracy for any fixed Reynolds number.
Abstract: We consider a two-level method for resolving the nonlinearity in finite element approximations of the equilibrium Navier--Stokes equations. The method yields L2 and H1 optimal velocity approximations and an L2 optimal pressure approximation. The two-level method involves solving one small, nonlinear coarse mesh system, one Oseen problem (hence, linear with positive definite symmetric part) on the fine mesh, and one linear correction problem on the coarse mesh. The algorithm we study produces an approximate solution with the optimal, asymptotic in h, accuracy for any fixed Reynolds number. We do not consider the behavior of the error for fixed h as $Re\rightarrow \infty$, i.e., for flows in transition to turbulence.
TL;DR: It is shown that, under some conditions of subcharacteristic type, for any convex entropy associated with the pressure p, the author can find a global convex and uniform entropy for the relaxation system.
Abstract: We consider the Euler equations for a compressible inviscid fluid with a general pressure law $p(\rho,\varepsilon)$, where $\rho$ represents the density of the fluid and $\varepsilon$ its specific internal energy. We show that it is possible to introduce a relaxation of the nonlinear pressure law introducing an energy decomposition under the form $\varepsilon= \varepsilon _1 + \varepsilon _2.$ The internal energy $\varepsilon _1$ is associated with a (simpler) pressure law $p_1(\rho,\varepsilon_1)$; the energy $\varepsilon _2$ is advected by the flow. These two energies are also subject to a relaxation process and in the limit of an infinite relaxation rate, we recover the initial pressure law p. We show that, under some conditions of subcharacteristic type, for any convex entropy associated with the pressure p, we can find a global convex and uniform entropy for the relaxation system.
From our construction, we also deduce the extension to general pressure laws of classical approximate Riemann solvers for polytropic gases, which only use a single call to the pressure law (per mesh point and time step). For the Godunov scheme, we show that this extension satisfies stability, entropy, and accuracy conditions.
TL;DR: The procedure is used to construct a new family of nonstaggered, central schemes for hyperbolic conservation laws by converting the family of staggered central schemes recently introduced, which retain the desirable properties of simplicity and high resolution and yield Riemann-solver-free recipes which avoid dimensional splitting.
Abstract: We present a general procedure to convert schemes which are based on staggered spatial grids into nonstaggered schemes. This procedure is then used to construct a new family of nonstaggered, central schemes for hyperbolic conservation laws by converting the family of staggered central schemes recently introduced in [H. Nessyahu and E. Tadmor, J. Comput. Phys., 87 (1990), pp. 408--463; X. D. Liu and E. Tadmor, Numer. Math., 79 (1998), pp. 397--425; G. S. Jiang and E. Tadmor, SIAM J. Sci. Comput., 19 (1998), pp. 1892--1917]. These new nonstaggered central schemes retain the desirable properties of simplicity and high resolution, and in particular, they yield Riemann-solver-free recipes which avoid dimensional splitting. Most important, the new central schemes avoid staggered grids and hence are simpler to implement in frameworks which involve complex geometries and boundary conditions.
TL;DR: In this paper, a two-level finite difference scheme for the approximation of nonlinear parabolic equations is presented, in which the full nonlinear problem is solved on a "coarse" grid of size H and an appropriate interpolation operator is used to provide values of the coarse grid solution on the fine grid in terms of superconvergent node points.
Abstract: We present a two-level finite difference scheme for the approximation of nonlinear parabolic equations. Discrete inner products and the lowest-order Raviart--Thomas approximating space are used in the expanded mixed method in order to develop the finite difference scheme. Analysis of the scheme is given assuming an implicit time discretization. In this two-level scheme, the full nonlinear problem is solved on a "coarse" grid of size H. The nonlinearities are expanded about the coarse grid solution and an appropriate interpolation operator is used to provide values of the coarse grid solution on the fine grid in terms of superconvergent node points. The resulting linear but nonsymmetric system is solved on a "fine" grid of size h. Some a priori error estimates are derived which show that the discrete L\infty(L2) and L2(H1) errors are $O(h^2 + H^{4-d/2} + \Delta t)$, where $d \geq 1$ is the spatial dimension.
TL;DR: In this article, the authors reformulate the problem in the form commonly used for the relaxation schemes to conservation laws by properly combining the stiff component of the convection terms into the relaxation term.
Abstract: Many kinetic models of the Boltzmann equation have a diffusive scaling that leads to the Navier--Stokes-type parabolic equations, such as the heat equation, the porous media equations, the advection-diffusion equation, and the viscous Burgers equation In such problems the diffusive relaxation parameter may differ in several orders of magnitude from the rarefied regimes to the hydrodynamic (diffusive) regimes, and it is desirable to develop a class of numerical schemes that can work uniformly with respect to this relaxation parameter Earlier approaches that work from the rarefied regimes to the Euler regimes do not directly apply to these problems since here, in addition to the stiff relaxation term, the convection term is also stiff Our idea is to reformulate the problem in the form commonly used for the relaxation schemes to conservation laws by properly combining the stiff component of the convection terms into the relaxation term This, however, introduces new difficulties due to the dependence of the stiff source term on the gradient We show how to overcome this new difficulty with an adequately designed, economical discretization procedure for the relaxation term These schemes are shown to have the correct diffusion limit Several numerical results in one and two dimensions are presented, which show the robustness, as well as the uniform accuracy, of our schemes
TL;DR: In this paper, an asymptotic-induced scheme for nonstationary transport equations with the diffusion scaling is developed, which works uniformly for all ranges of mean-free paths.
Abstract: An asymptotic-induced scheme for nonstationary transport equations with the diffusion scaling is developed. The scheme works uniformly for all ranges of mean-free paths. It is based on the asymptotic analysis of the diffusion limit of the transport equation.
A theoretical investigation of the behavior of the scheme in the diffusion limit is given and an approximation property is proven. Moreover, numerical results for different physical situations are shown and the uniform convergence of the scheme is established numerically.
TL;DR: It is proved then that a careful coupling of time and space discretizations can allow large time steps in the numerical integration still preserving the accuracy of the solutions.
Abstract: The convergence properties of a class of high-order semi-Lagrangian schemes for pure advection equations are studied here in the framework of the theory of viscosity solutions. We review the general convergence results for discrete-time approximation schemes belonging to that class and we prove some a priori estimates in $L^\infty$ and L2 for the rate of convergence of fully discrete schemes. We prove then that a careful coupling of time and space discretizations can allow large time steps in the numerical integration still preserving the accuracy of the solutions. Several examples of schemes and numerical tests are presented.
TL;DR: A generalization of steepest descent and other methods for solving a large scale symmetric positive definitive system Ax = b is presented in this paper, where the new iteration is given by $x_{k+1} = x_k - \lambda (x_{
u(k)}) (A xk - b)$, where the steepest descend step at a previous iteration is the one at which the solution was chosen.
Abstract: A generalization of the steepest descent and other methods for solving a large scale symmetric positive definitive system Ax = b is presented. Given a positive integer m, the new iteration is given by $x_{k+1} = x_k - \lambda (x_{
u(k)}) (A x_k - b)$, where $\lambda (x_{
u(k)})$ is the steepest descent step at a previous iteration $
u(k) \in \{k, k-1, \ldots,$ max$\{0,k-m\}\}$. The global convergence to the solution of the problem is established under a more general framework, and numerical experiments are performed that suggest that some strategies for the choice of $
u(k)$ give rise to efficient methods for obtaining approximate solutions of the system.
TL;DR: In this article, a method to prove the existence and the local uniqueness of solutions to infinite-dimensional fixed-point equations using computers was proposed, where the set which possibly includes a solution is chosen by an approximate linearization of the operator appearing in the equation.
Abstract: In this paper, we propose a method to prove the existence and the local uniqueness of solutions to infinite-dimensional fixed-point equations using computers. Choosing a set which possibly includes a solution, we transform it by an approximate linearization of the operator appearing in the equation. Then we calculate the radii of the transformed set in order to check sufficient conditions for Banach's fixed-point theorem. This method is applied to elliptic problems and numerical examples are given.
TL;DR: In this paper, a set of tools for verifying smoothness of surfaces generated by stationary subdivision algorithms is presented, where the main challenge is the verification of injectivity of the characteristic map.
Abstract: A new set of tools for verifying smoothness of surfaces generated by stationary subdivision algorithms is presented The main challenge here is the verification of injectivity of the characteristic map The tools are sufficiently versatile and easy to wield to allow, as an application, a full analysis of algorithms generalizing biquadratic and bicubic B-spline subdivision In the case of generalized biquadratic subdivision the analysis yields a hitherto unknown sharp bound strictly less than 1 on the second largest eigenvalue of any smoothly converging subdivision
TL;DR: The authors provide a counterexample to show that an expected L2-error estimate does not exist in the usual sense and it is conjectured that the optimal order of $\|u-u_h\|_{0,\Omega}$ should be O(h) for the general case.
Abstract: The finite volume element method (FVE) is a discretization technique for partial differential equations. This paper develops discretization energy error estimates for general self-adjoint elliptic boundary value problems with FVE based on triangulations, on which there exist linear finite element spaces, and a very general type of control volumes (covolumes).
The energy error estimates of this paper are also optimal but the restriction conditions for the covolumes given in [R. E. Bank and D. J. Rose, SIAM J. Numer. Anal., 24 (1987), pp. 777--787], [Z. Q. Cai, Numer. Math., 58 (1991), pp. 713--735] are removed. The authors finally provide a counterexample to show that an expected L2-error estimate does not exist in the usual sense. It is conjectured that the optimal order of $\|u-u_h\|_{0,\Omega}$ should be O(h) for the general case.
TL;DR: In this paper, a new version of Lax-Friedrichs and an associated second-order predictor-corrector method are presented for scalar advection in two dimensions.
Abstract: Global composition of several time steps of the two-step Lax--Wendroff scheme followed by a Lax--Friedrichs step seems to enhance the best features of both, although it is only first order accurate. We show this by means of some examples of one-dimensional shallow water flow over an obstacle. In two dimensions we present a new version of Lax--Friedrichs and an associated second order predictor-corrector method. Composition of these schemes is shown to be effective and efficient for some two-dimensional Riemann problems and for Noh's infinite strength cylindrical shock problem. We also show comparable results for composition of the predictor-corrector scheme with a modified second order accurate weighted essentially nonoscillatory (WENO) scheme. That composition is second order but is more efficient and has better symmetry properties than WENO alone. For scalar advection in two dimensions the optimal stability of the new predictor-corrector scheme is shown using computer algebra. We also show that the generalization of this scheme to three dimensions is unstable, but by using sampling we are able to show that the composites are suboptimally stable.
TL;DR: In this article, an H1-Galerkin mixed finite element method is proposed and analyzed for parabolic partial differential equations with nonselfadjoint elliptic parts, and it is shown that the finite element approximations have the same rates of convergence as in the classical mixed method.
Abstract: In this paper, an H1-Galerkin mixed finite element method is proposed and analyzed for parabolic partial differential equations with nonselfadjoint elliptic parts. Compared to the standard H1-Galerkin procedure, C1-continuity for the approximating finite dimensional subspaces can be relaxed for the proposed method. Moreover, it is shown that the finite element approximations have the same rates of convergence as in the classical mixed method, but without LBB consistency condition and quasiuniformity requirement on the finite element mesh. Finally, a better rate of convergence for the flux in L2-norm is derived using a modified H1-Galerkin mixed method in two and three space dimensions, which confirms the findings in a single space variable and also improves upon the order of convergence of the classical mixed method under extra regularity assumptions on the exact solution.
TL;DR: In this article, the Landau-de Gennes free energy minimization problem for computing equilibrium configurations of the tensor order parameter field that characterizes the molecular orientational properties of liquid crystal materials is addressed.
Abstract: This paper describes the Landau--de Gennes free-energy minimization problem for computing equilibrium configurations of the tensor order parameter field that characterizes the molecular orientational properties of liquid crystal materials. Analytical and numerical issues are addressed. Conditions guaranteeing well-posedness (existence, regularity) of the problem are given, as is a nonlinear finite element convergence analysis.
TL;DR: In this paper, a numerical homogenization procedure for elliptic differential equations is presented, based on wavelet decompositions of discrete operators in fine and coarse scale components followed by the elimination of the fine scale contributions.
Abstract: A numerical homogenization procedure for elliptic differential equations is presented. It is based on wavelet decompositions of discrete operators in fine and coarse scale components followed by the elimination of the fine scale contributions. If the operator is in divergence form, this is preserved by the homogenization procedure. For periodic problems, results similar to classical effective coefficient theory are proved. The procedure can be applied to problems that are not cell-periodic.
TL;DR: In this article, the authors studied the numerical transition from a Hamilton-Jacobi (H--J) equation to its associated system of conservation laws in arbitrary space dimensions and introduced a simple second-order relaxation scheme to solve the underlying weakly hyperbolic system.
Abstract: In this paper we study the numerical transition from a Hamilton--Jacobi (H--J) equation to its associated system of conservation laws in arbitrary space dimensions. We first study how, in a very generic setting, one can recover the viscosity solutions of the H--J equation using the numerical solutions to the system of conservation laws. We then introduce a simple, second-order relaxation scheme to solve the underlying weakly hyperbolic system of conservation laws.
TL;DR: In this article, a least-squares approach based on L2 norms was proposed to solve the Navier-Stokes equations in primitive variables. But this approach does not allow practical implementation, and these results are critical to the analysis of a practical least square method for the reduced system based on a discrete equivalent of the negative norm.
Abstract: This paper develops a least-squares approach to the solution of the incompressible Navier--Stokes equations in primitive variables. As with our earlier work on Stokes equations, we recast the Navier--Stokes equations as a first-order system by introducing a velocity-flux variable and associated curl and trace equations. We show that a least-squares principle based on L2 norms applied to this system yields optimal discretization error estimates in the H1 norm in each variable, including the velocity flux.
An analogous principle based on the use of an H-1 norm for the reduced system (with no curl or trace constraints) is shown to yield similar estimates, but now in the L2 norm for velocity-flux and pressure. Although the H-1 least-squares principle does not allow practical implementation, these results are critical to the analysis of a practical least-squares method for the reduced system based on a discrete equivalent of the negative norm. A practical method of this type is the subject of a companion paper. Finally, we establish optimal multigrid convergence estimates for the algebraic system resulting from the L2 norm approach.
TL;DR: This paper introduces a general technique for the construction of multistep methods capable of integrating, without local truncation error, homogeneous linear ODEs with constant coefficients, including those, in particular, that result in oscillatory solutions.
Abstract: This paper introduces a general technique for the construction of multistep methods capable of integrating, without local truncation error, homogeneous linear ODEs with constant coefficients, including those, in particular, that result in oscillatory solutions. Moreover, these methods can be further adapted through coefficient modification for the exact integration of forced oscillations in one or more frequencies, even confluent ones that occur from nonhomogeneous terms in the differential equation. Our procedure allows the derivation of many of the existing codes with similar properties, as well as the improvement of others that in their original design were only able to integrate oscillations in a single frequency. The properties of the methods are studied within a general framework, and numerical examples are presented. These demonstrate the way in which the new algorithms perform distinctly better than the general purpose codes, particularly when integrating the class of equations with perturbed oscillatory solutions. The methods developed are mainly applicable to the accurate and efficient integration of problems for which the oscillation frequencies are known, as occurs in satellite orbit propagation. The underlying ideas have already been applied to the improvement of some Chebyshev methods that are not multistep.
TL;DR: In this article, it was shown that the local error may effect the smooth part of the solution such that only first order is achieved even for higher-order methods, and that a converging method will have the formal order of accuracy in domains where no characteristics have passed through a shock.
Abstract: Finite difference approximations generically have ${\cal O}(1)$ pointwise errors close to a shock. We show that this local error may effect the smooth part of the solution such that only first order is achieved even for formally higher-order methods. Analytic and numerical examples of this form of accuracy are given. We also show that a converging method will have the formal order of accuracy in domains where no characteristics have passed through a shock.
TL;DR: The postprocessed Galerkin method is shown, in the case of Fourier modes, to possess the same rate of convergence (accuracy) as the simplest version of the NLG, which is based on either the Foias--Manley--Temam approximate inertial manifold or the Euler--Galerkin approximate inertIAL manifold.
Abstract: A postprocess of the standard Galerkin method for the discretization of dissipative equations is presented. The postprocessed Galerkin method uses the same approximate inertial manifold $\Phi_{app}$ to approximate the high wave number modes of the solution as in the nonlinear Galerkin (NLG) method. However, in this postprocessed Galerkin method the value of $\Phi_{app}$ is calculated only once, after the time integration of the standard Galerkin method is completed, contrary to the NLG in which $\Phi_{app}$ evolves with time and affects the time evolution of the lower wave number modes. The postprocessed Galerkin method, which is much cheaper to implement computationally than the NLG, is shown, in the case of Fourier modes, to possess the same rate of convergence (accuracy) as the simplest version of the NLG, which is based on either the Foias--Manley--Temam approximate inertial manifold or the Euler--Galerkin approximate inertial manifold. This is proved for some problems in one and two spatial dimensions, including the Navier--Stokes equations under periodic boundary conditions. The advantages of postprocessing that we present here apply not only to the standard Galerkin method, but also to the computationally more efficient pseudospectral method.