Showing papers in "Numerische Mathematik in 2001"

Galerkin proper orthogonal decomposition methods for parabolic problems

Abstract: In this work error estimates for Galerkin proper orthogonal decomposition (POD) methods for linear and certain non-linear parabolic systems are proved. The resulting error bounds depend on the number of POD basis functions and on the time discretization. Numerical examples are included.

A stable and conservative finite difference scheme for the Cahn-Hilliard equation

Abstract: We propose a stable and conservative finite difference scheme to solve numerically the Cahn-Hilliard equation which describes a phase separation phenomenon. Numerical solutions to the equation is hard to obtain because it is a nonlinear and nearly ill-posed problem. We design a new difference scheme based on a general strategy proposed recently by Furihata and Mori. The new scheme inherits characteristic properties, the conservation of mass and the decrease of the total energy, from the equation. The decrease of the total energy implies boundedness of discretized Sobolev norm of the solution. This in turn implies, by discretized Sobolev's lemma, boundedness of max norm of the solution, and hence the stability of the solution. An error estimate for the solution is obtained and the order is \(O( (\Delta x)^2 + (\Delta t)^2)\). Numerical examples demonstrate the effectiveness of the proposed scheme.

Finite element discretization of the Navier–Stokes equations with a free capillary surface

Abstract: The instationary Navier–Stokes equations with a free capillary boundary are considered in 2 and 3 space dimensions. A stable finite element discretization is presented. The key idea is the treatment of the curvature terms by a variational formulation. In the context of a discontinuous in time space–time element discretization stability in (weak) energy norms can be proved. Numerical examples in 2 and 3 space dimensions are given.

New anisotropic a priori error estimates

Abstract: We prove a priori anisotropic estimates for the $L^2$ and $H^1$ interpolation error on linear finite elements. The full information about the mapping from a reference element is employed to separate the contribution to the elemental error coming from different directions. This new $H^1$ error estimate does not require the “maximal angle condition”. The analysis has been carried out for the 2D case, but may be extended to three dimensions. Numerical experiments have been carried out to test our theoretical results.

Second order Chebyshev methods based on orthogonal polynomials

Abstract: Stabilized methods (also called Chebyshev methods) are explicit Runge-Kutta methods with extended stability domains along the negative real axis. These methods are intended for large mildly stiff problems, originating mainly from parabolic PDEs. The aim of this paper is to show that with the use of orthogonal polynomials, we can construct nearly optimal stability polynomials of second order with a three-term recurrence relation. These polynomials can be used to construct a new numerical method, which is implemented in a code called ROCK2. This new numerical method can be seen as a combination of van der Houwen-Sommeijer-type methods and Lebedev-type methods.

Discrete Hodge operators

Abstract: Many linear boundary value problems arising in computational physics can be formulated in the calculus of differential forms. Discrete differential forms provide a natural and canonical approach to their discretization. However, much freedom remains concerning the choice of discrete Hodge operators, that is, discrete analogues of constitutive laws. A generic discrete Hodge operator is introduced and it turns out that most finite element and finite volume schemes emerge as its specializations. We reap the possibility of a unified convergence analysis in the framework of discrete exterior calculus.

A third-order semi-discrete genuinely multidimensional central scheme for hyperbolic conservation laws and related problems

Abstract: We construct a new third-order semi-discrete genuinely multidimensional central scheme for systems of conservation laws and related convection-diffusion equations. This construction is based on a multidimensional extension of the idea, introduced in [17] – the use of more precise information about the local speeds of propagation, and integration over nonuniform control volumes, which contain Riemann fans. As in the one-dimensional case, the small numerical dissipation, which is independent of ${\cal O}(\frac{1}{\Delta t})$ , allows us to pass to a limit as $\Delta t \downarrow 0$ . This results in a particularly simple genuinely multidimensional semi-discrete scheme. The high resolution of the proposed scheme is ensured by the new two-dimensional piecewise quadratic non-oscillatory reconstruction. First, we introduce a less dissipative modification of the reconstruction, proposed in [29]. Then, we generalize it for the computation of the two-dimensional numerical fluxes. Our scheme enjoys the main advantage of the Godunov-type central schemes –simplicity, namely it does not employ Riemann solvers and characteristic decomposition. This makes it a universal method, which can be easily implemented to a wide variety of problems. In this paper, the developed scheme is applied to the Euler equations of gas dynamics, a convection-diffusion equation with strongly degenerate diffusion, the incompressible Euler and Navier-Stokes equations. These numerical experiments demonstrate the desired accuracy and high resolution of our scheme.

Crouzeix-Raviart type finite elements on anisotropic meshes

Abstract: The paper deals with a non-conforming finite element method on a class of anisotropic meshes. The Crouzeix-Raviart element is used on triangles and tetrahedra. For rectangles and prismatic (pentahedral) elements a novel set of trial functions is proposed. Anisotropic local interpolation error estimates are derived for all these types of element and for functions from classical and weighted Sobolev spaces. The consistency error is estimated for a general differential equation under weak regularity assumptions. As a particular application, an example is investigated where anisotropic finite element meshes are appropriate, namely the Poisson problem in domains with edges. A numerical test is described.

$L^1$-Stability and error estimates for approximate Hamilton-Jacobi solutions

Abstract: We study the $L^1$ -stability and error estimates of general approximate solutions for the Cauchy problem associated with multidimensional Hamilton-Jacobi (H-J) equations. For strictly convex Hamiltonians, we obtain a priori error estimates in terms of the truncation errors and the initial perturbation errors. We then demonstrate this general theory for two types of approximations: approximate solutions constructed by the vanishing viscosity method, and by Godunov-type finite difference methods. If we let $\epsilon$ denote the `small scale' of such approximations (– the viscosity amplitude $\epsilon$ , the spatial grad-size $\Delta x$ , etc.), then our $L^1$ -error estimates are of ${\cal O}(\epsilon)$ , and are sharper than the classical $L^\infty$ -results of order one half, ${\cal O}(\sqrt{\epsilon})$ . The main building blocks of our theory are the notions of the semi-concave stability condition and $L^1$ -measure of the truncation error. The whole theory could be viewed as a multidimensional extension of the $Lip^\prime$ -stability theory for one-dimensional nonlinear conservation laws developed by Tadmor et. al. [34,24,25]. In addition, we construct new Godunov-type schemes for H-J equations which consist of an exact evolution operator and a global projection operator. Here, we restrict our attention to linear projection operators (first-order schemes). We note, however, that our convergence theory applies equally well to nonlinear projections used in the context of modern high-resolution conservation laws. We prove semi-concave stability and obtain $L^1$ -bounds on their associated truncation errors; $L^1$ -convergence of order one then follows. Second-order (central) Godunov-type schemes are also constructed. Numerical experiments are performed; errors and orders are calculated to confirm our $L^1$ -theory.

Algebraic theory of multiplicative Schwarz methods

Abstract: Summary. The convergence of multiplicative Schwarz-type methods for solving linear systems when the coefficient matrix is either a nonsingular M -matrix or a symmetric positive definite matrix is studied using classical and new results from the theory of splittings. The effect on convergence of algorithmic parameters such as the number of subdomains, the amount of overlap, the result of inexact local solves and of “coarse grid” corrections (global coarse solves) is analyzed in an algebraic setting. Results on algebraic additive Schwarz are also included.

A posteriori error estimate and h-adaptive algorithm on surfaces for Symm's integral equation

Abstract: A residual-based a posteriori error estimate for boundary integral equations on surfaces is derived in this paper. A localisation argument involves a Lipschitz partition of unity such as nodal basis functions known from finite element methods. The abstract estimate does not use any property of the discrete solution, but simplifies for the Galerkin discretisation of Symm's integral equation if piecewise constants belong to the test space. The estimate suggests an isotropic adaptive algorithm for automatic mesh-refinement. An alternative motivation from a two-level error estimate is possible but then requires a saturation assumption. The efficiency of an anisotropic version is discussed and supported by numerical experiments.

Computing the confluent hypergeometric function, M(a,b,x)

Abstract: The confluent hypergeometric function, M(a,b,x), arises naturally in both statistics and physics. Although analytically well-behaved, extreme but practically useful combinations of parameters create extreme computational difficulties. A brief review of known analytic and computational results highlights some difficult regions, including $b>a>0$ , with x much larger than b. Existing power series and integral representations may fail to converge numerically, while asymptotic series representations may diverge before achieving the accuracy desired. Continued fraction representations help somewhat. Variable precision can circumvent the problem, but with reductions in speed and convenience. In some cases, known analytic properties allow transforming a difficult computation into an easier one. The combination of existing computational forms and transformations still leaves gaps. For $b>a>0$ , two new power series, in terms of Gamma and Beta cumulative distribution functions respectively, help in some cases. Numerical evaluations highlight the abilities and limitations of existing and new methods. Overall, a rational approximation due to Luke and the new Gamma-based series provide the best performance.

Multiresolution schemes for conservation laws

Abstract: The concept of fully adaptive multiresolution finite volume schemes has been developed and investigated during the past decade. By now it has been successfully employed in numerous applications arising in engineering. In the present work a review on the methodology is given that aims to summarize the underlying concepts and to give an outlook on future developments.

Semi-implicit finite volume scheme for solving nonlinear diffusion equations in image processing

Abstract: We propose and prove a convergence of the semi-implicit finite volume approximation scheme for the numerical solution of the modified (in the sense of Catte, Lions, Morel and Coll) Perona–Malik nonlinear image selective smoothing equation (called anisotropic diffusion in the image processing). The proof is based on $L_2$ a-priori estimates and Kolmogorov's compactness theorem. The implementation aspects and computational results are discussed.

On convergence rates of inexact Newton regularizations

Abstract: REGINN is an algorithm of inexact Newton type for the regularization of nonlinear ill-posed problems [Inverse Problems 15 (1999), pp. 309–327]. In the present article convergence is shown under weak smoothness assumptions (source conditions). Moreover, convergence rates are established. Some computational illustrations support the theoretical results.

New properties of a nonlinear conjugate gradient method

Abstract: This paper provides several new properties of the nonlinear conjugate gradient method in [5]. Firstly, the method is proved to have a certain self-adjusting property that is independent of the line search and the function convexity. Secondly, under mild assumptions on the objective function, the method is shown to be globally convergent with a variety of line searches. Thirdly, we find that instead of the negative gradient direction, the search direction defined by the nonlinear conjugate gradient method in [5] can be used to restart any optimization method while guaranteeing the global convergence of the method. Some numerical results are also presented.

Higher order numerical schemes for affinely controlled nonlinear systems

Abstract: A systematic method for the derivation of high order schemes for affinely controlled nonlinear systems is developed. Using an adaptation of the stochastic Taylor expansion for control systems we construct Taylor schemes of arbitrary high order and indicate how derivative free Runge-Kutta type schemes can be obtained. Furthermore an approximation technique for the multiple control integrals appearing in the schemes is proposed.

On the zero-stability of variable stepsize multistep methods: the spectral radius approach

Abstract: In this paper we illustrate a novel approach for studying the asymptotic behaviour of the solutions of linear difference equations with variable coefficients. In particular, we deal with the zero-stability of the 3-step BDF-method on grids with variable stepsize for the numerical solution of IVPs for ODEs. Our approach is based on the theory of the spectral radius of a family of matrices and yields almost optimal results, which give a slight improvement to the best results already known from the literature. The success got on the chosen example suggests that our approach has a good potential for more general and harder stability analyses of numerical methods.

Additive Schwarz method for the p-version of the boundary element method for the single layer potential operator on a plane screen

Abstract: We analyze an additive Schwarz preconditioner for the p-version of the boundary element method for the single layer potential operator on a plane screen in the three-dimensional Euclidean space. We decompose the ansatz space, which consists of piecewise polynomials of degree p on a mesh of size h, by introducing a coarse mesh of size $H\ge h$ . After subtraction of the coarse subspace of piecewise constant functions on the coarse mesh this results in local subspaces of piecewise polynomials living only on elements of size H. This decomposition yields a preconditioner which bounds the spectral condition number of the stiffness matrix by $O(\log \frac Hhp)^2$ . Numerical results supporting the theory are presented.

Analysis of a coupled finite-infinite element method for exterior Helmholtz problems

Abstract: This analysis of convergence of a coupled FEM-IEM is based on our previous work on the FEM and the IEM for exterior Helmholtz problems. The key idea is to represent both the exact and the numerical solution by the Dirichlet-to-Neumann operators that they induce on the coupling hypersurface in the exterior of an obstacle. The investigation of convergence can then be related to a spectral analysis of these DtN operators. We give a general outline of our method and then proceed to a detailed investigation of the case that the coupling surface is a sphere. Our main goal is to explore the convergence mechanism. In this context, we show well-posedness of both the continuous and the discrete models. We further show that the discrete inf-sup constants have a positive lower bound that does not depend on the number of DOF of the IEM. The proofs are based on lemmas on the spectra of the continuous and the discrete DtN operators, where the spectral characterization of the discrete DtN operator is given as a conjecture from numerical experiments. In our convergence analysis, we show algebraic (in terms of N) convergence of arbitrary order and generalize this result to exponential convergence.

A posteriori error estimate for finite volume approximations to singularly perturbed nonlinear convection-diffusion equations

Abstract: This paper is devoted to the study of a posteriori and a priori error estimates for the scalar nonlinear convection diffusion equation $c_t + abla \cdot ( \u f(c)) - \varepsilon \Delta c = 0$ . The estimates for the error between the exact solution and an upwind finite volume approximation to the solution are derived in the $L^1$ -norm in the situation, where the diffusion parameter $\varepsilon$ is smaller or comparable to the mesh size. Numerical experiments underline the theoretical results.

Strictly feasible equation-based methods for mixed complementarity problems

Abstract: We introduce a new algorithm for the solution of the mixed complementarity problem (MCP) which has stronger properties than most existing methods. In fact, typical solution methods for the MCP either generate feasible iterates but have to solve relatively complicated subproblems (like quadratic programs or linear complementarity problems), or they have relatively simple subproblems (like linear systems of equations) but generate not necessarily feasible iterates. The method to be presented here combines the nice features of these two classes of methods: It has to solve only one linear system of equations (of reduced dimension) at each iteration, and it generates feasible (more precisely: strictly feasible) iterates. The new method has some nice global and local convergence properties. Some preliminary numerical results will also be given.

Heterogeneous coupling by virtual control methods

Abstract: The virtual control method, recently introduced to approximate elliptic and parabolic problems by overlapping domain decompositions (see [7–9]), is proposed here for heterogeneous problems. Precisely, we address the coupling of an advection equation with a diffusion-advection equation, with the aim of modelling boundary layers. We investigate both overlapping and non-overlapping (disjoint) subdomain decompositions. In the latter case, several cost functions are considered and a numerical assessment of our theoretical conclusions is carried out.

Integration methods based on canonical coordinates of the second kind

Abstract: We present a new class of integration methods for differential equations on manifolds, in the framework of Lie group actions. Canonical coordinates of the second kind is used for representing the Lie group locally by means of its corresponding Lie algebra. The coordinate map itself can, in many cases, be computed inexpensively, but the approach also involves the inversion of its differential, a task that can be challenging. To succeed, it is necessary to consider carefully how to choose a basis for the Lie algebra, and the ordering of the basis is important as well. For semisimple Lie algebras, one may take advantage of the root space decomposition to provide a basis with desirable properties. The problem of ordering leads us to introduce the concept of an admissible ordered basis (AOB). The existence of an AOB is established for some of the most important Lie algebras. The computational cost analysis shows that the approach may lead to more efficient solvers for ODEs on manifolds than those based on canonical coordinates of the first kind presented by Munthe-Kaas. Numerical experiments verify the derived properties of the new methods.

The sdfem on Shishkin meshes for linear convection-diffusion problems

Abstract: Summary. We consider singularly perturbed linear elliptic problems in two dimensions. The solutions of such problems typically exhibit layers and are difficult to solve numerically. The streamline diffusion finite element method (SDFEM) has been proved to produce accurate solutions away from any layers on uniform meshes, but fails to compute the boundary layers precisely. Our modified SDFEM is implemented with piecewise linear functions on a Shishkin mesh that resolves boundary layers, and we prove that it yields an accurate approximation of the solution both inside and outside these layers. The analysis is complicated by the severe nonuniformity of the mesh. We give local $L_\infty$ error estimates that hold true uniformly in the perturbation parameter $\varepsilon$ , provided only that $\varepsilon \le N^{-1}$ , where ${\cal O}(N^2)$ mesh points are used. Numerical experiments support these theoretical results.

Convergence of a relaxation scheme for hyperbolic systems of conservation laws

Abstract: This paper concerns the study of a relaxation scheme for $N\times N$ hyperbolic systems of conservation laws. In particular, with the compensated compactness techniques, we prove a rigorous result of convergence of the approximate solutions toward an entropy solution of the equilibrium system, as the relaxation time and the mesh size tend to zero.

Macro-elements and stable local bases for splines on Clough-Tocher triangulations

Abstract: Macro-elements of arbitrary smoothness are constructed on Clough-Tocher triangle splits. These elements can be used for solving boundary-value problems or for interpolation of Hermite data, and are shown to be optimal with respect to spline degree. We conjecture they are also optimal with respect to the number of degrees of freedom. The construction provides local bases for certain superspline spaces defined over Clough-Tocher refinements of arbitrary triangulations. These bases are shown to be stable as a function of the smallest angle in the triangulation, which in turn implies that the associated spline spaces have optimal order approximation power.

An adaptive discretization for Tikhonov-Phillips regularization with a posteriori parameter selection

Abstract: Summary. The aim of this paper is to describe an efficient adaptive strategy for discretizing ill-posed linear operator equations of the first kind: we consider Tikhonov-Phillips regularization \[ x_{\alpha}^{\delta} = \left(A^{\ast}A+\alpha I\right)^{-1}A^{\ast}y^{\delta} \] with a finite dimensional approximation $A_n$ instead of A. We propose a sparse matrix structure which still leads to optimal convergences rates but requires substantially less scalar products for computing $A_n$ compared with standard methods.

Lagrangian systems of conservation laws

Abstract: We study the mathematical structure of 1D systems of conservation laws written in the Lagrange variable. Modifying the symmetrization proof of systems of conservation laws with three hypothesis, we prove that these models have a canonical formalism. These hypothesis are i) the entropy flux is zero, ii) Galilean invariance, iii) reversibility for smooth solutions. Then we study a family of numerical schemes for the solution of these systems. We prove that they are entropy consistent. We also prove from general considerations the symmetry of the spectrum of the Jacobian matrix.

Degenerate two-phase incompressible flow: III. Sharp error estimates

Abstract: This is the third paper of a series in which we analyze mathematical properties and develop numerical methods for a degenerate elliptic-parabolic partial differential system which describes the flow of two incompressible, immiscible fluids in porous media In this paper we consider a finite element approximation for this system The elliptic equation for the pressure and velocity is approximated by a mixed finite element method, while the degenerate parabolic equation for the saturation is approximated by a Galerkin finite element method A fully discrete approximation is analyzed Sharp error estimates in energy norms are obtained for this approximation The error analysis does not use any regularization of the saturation equation; the error estimates are derived directly from the degenerate equation Also, the analysis does not impose any restriction on the nature of degeneracy Finally, it respects the minimal regularity on the solution of the differential system