Showing papers in "Numerische Mathematik in 2004"
TL;DR: In this paper, the adaptive finite element method for solving the Laplace equation with piecewise linear elements on domains in ℝ2 was proposed and proved to have a convergence rate of O(n−s) in the energy norm.
Abstract: Adaptive Finite Element Methods for numerically solving elliptic equations are used often in practice. Only recently [12], [17] have these methods been shown to converge. However, this convergence analysis says nothing about the rates of convergence of these methods and therefore does, in principle, not guarantee yet any numerical advantages of adaptive strategies versus non-adaptive strategies. The present paper modifies the adaptive method of Morin, Nochetto, and Siebert [17] for solving the Laplace equation with piecewise linear elements on domains in ℝ2 by adding a coarsening step and proves that this new method has certain optimal convergence rates in the energy norm (which is equivalent to the H1 norm). Namely, it is shown that whenever s>0 and the solution u is such that for each n≥1, it can be approximated to accuracy O(n−s) in the energy norm by a continuous, piecewise linear function on a triangulation with n cells (using complete knowledge of u), then the adaptive algorithm constructs an approximation of the same type with the same asymptotic accuracy while using only information gained during the computational process. Moreover, the number of arithmetic computations in the proposed method is also of order O(n) for each n≥1. The construction and analysis of this adaptive method relies on the theory of nonlinear approximation.
564 citations
TL;DR: A class of preconditioned Hermitian/skew-Hermitian splitting iteration methods is established, showing that the new method converges unconditionally to the unique solution of the linear system.
Abstract: For the positive semidefinite system of linear equations of a block two-by-two structure, by making use of the Hermitian/skew-Hermitian splitting iteration technique we establish a class of preconditioned Hermitian/skew-Hermitian splitting iteration methods. Theoretical analysis shows that the new method converges unconditionally to the unique solution of the linear system. Moreover, the optimal choice of the involved iteration parameter and the corresponding asymptotic convergence rate are computed exactly. Numerical examples further confirm the correctness of the theory and the effectiveness of the method.
465 citations
TL;DR: An implicit discretization method is developed for pricing such American options where the underlying asset follows a jump diffusion process and sufficient conditions for global convergence of the discrete penalized equations at each timestep are derived.
Abstract: The fair price for an American option where the underlying asset follows a jump diffusion process can be formulated as a partial integral differential linear complementarity problem. We develop an implicit discretization method for pricing such American options. The jump diffusion correlation integral term is computed using an iterative method coupled with an FFT while the American constraint is imposed by using a penalty method. We derive sufficient conditions for global convergence of the discrete penalized equations at each timestep. Finally, we present numerical tests which illustrate such convergence.
226 citations
TL;DR: A new mixed variational formulation of the equations of stationary incompressible magneto–hydrodynamics is introduced and analyzed, based on curl-conforming Sobolev spaces for the magnetic variables and shown to be well-posed in (possibly non-convex) Lipschitz polyhedra.
Abstract: A new mixed variational formulation of the equations of stationary incompressible magneto–hydrodynamics is introduced and analyzed. The formulation is based on curl-conforming Sobolev spaces for the magnetic variables and is shown to be well-posed in (possibly non-convex) Lipschitz polyhedra. A finite element approximation is proposed where the hydrodynamic unknowns are discretized by standard inf-sup stable velocity-pressure space pairs and the magnetic ones by a mixed approach using Nedelec’s elements of the first kind. An error analysis is carried out that shows that the proposed finite element approximation leads to quasi-optimal error bounds in the mesh-size.
188 citations
TL;DR: It is shown that all error bounds depend on only in some lower polynomial order for small ɛ, and convergence of the fully discrete finite element solution to the solution of the Hele-Shaw (Mullins-Sekerka) problem is proved.
Abstract: We propose and analyze a semi-discrete and a fully discrete mixed finite element method for the Cahn-Hilliard equation ut + Δ(ɛΔu−ɛ−1f(u)) = 0, where ɛ > 0 is a small parameter. Error estimates which are quasi-optimal order in time and optimal order in space are shown for the proposed methods under minimum regularity assumptions on the initial data and the domain. In particular, it is shown that all error bounds depend on ** only in some lower polynomial order for small ɛ. The cruxes of our analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of Alikakos and Fusco [2], and Chen [15] to prove a discrete counterpart of it for a linearized Cahn-Hilliard operator to handle the nonlinear term on a stretched time grid. The ideas and techniques developed in this paper also enable us to prove convergence of the fully discrete finite element solution to the solution of the Hele-Shaw (Mullins-Sekerka) problem as ɛ → 0 in [29].
164 citations
TL;DR: A Kružkov-type notion of entropy solution is suggested to be used for this scalar conservation law modeling the settling of particles in an ideal clarifier-thickener unit to prove uniqueness (L1 stability) of the entropy solution in the BVt class.
Abstract: We consider a scalar conservation law modeling the settling of particles in an ideal clarifier-thickener unit. The conservation law has a nonconvex flux which is spatially dependent on two discontinuous parameters. We suggest to use a Kružkov-type notion of entropy solution for this conservation law and prove uniqueness (L1 stability) of the entropy solution in the BVt class (functions W(x,t) with ∂tW being a finite measure). The existence of a BVt entropy solution is established by proving convergence of a simple upwind finite difference scheme (of the Engquist-Osher type). A few numerical examples are also presented.
121 citations
TL;DR: It is shown that it is possible to find near optimal trees using computations linear in n and the best tree based on n adaptive decisions could be found by examining all such possibilities.
Abstract: Adaptive methods of approximation arise in many settings including numerical methods for PDEs and image processing. They can usually be described by a tree which records the adaptive decisions. This paper is concerned with the fast computation of near optimal trees based on n adaptive decisions. The best tree based on n adaptive decisions could be found by examining all such possibilities. However, this is exponential in n and could be numerically prohibitive. The main result of this paper is to show that it is possible to find near optimal trees using computations linear in n.
112 citations
TL;DR: Convergence is proven for monotone schemes and numerical tests are presented and discussed and the numerical approximation of viscosity solutions for integro-differential, possibly degenerate, parabolic problems is studied.
Abstract: We study the numerical approximation of viscosity solutions for integro-differential, possibly degenerate, parabolic problems. Similar models arise in option pricing, to generalize the celebrated Black–Scholes equation, when the processes which generate the underlying stock returns may contain both a continuous part and jumps. Convergence is proven for monotone schemes and numerical tests are presented and discussed.
103 citations
TL;DR: In addition to showing well-posedness of the approximation, this work proves convergence in space dimensions of the sixth order nonlinear degenerate parabolic equation.
Abstract: We consider a finite element approximation of the sixth order nonlinear degenerate parabolic equation **** equation here *** where generically **** equation here *** for any given **** equation here *** In addition to showing well-posedness of our approximation, we prove convergence in space dimensions $d \leq 3$. Furthermore an iterative scheme for solving the resulting nonlinear discrete system is analysed. Finally some numerical experiments in one and two space dimensions are presented.
103 citations
TL;DR: An a posteriori error estimator for parameter identification problems using the general setting of a partial differential equation written in weak form with abstract parameter dependence to derive an error estimators which is cheap in comparison to the overall optimization algorithm.
Abstract: In this paper we develop an a posteriori error estimator for parameter identification problems. The state equation is given by a partial differential equation involving a finite number of unknown parameters. The presented error estimator aims to control the error in the parameters due to discretization by finite elements. For this, we consider the general setting of a partial differential equation written in weak form with abstract parameter dependence. Exploiting the special structure of the parameter identification problem, allows us to derive an error estimator which is cheap in comparison to the overall optimization algorithm. Several examples illustrating the behavior of an adaptive mesh refinement algorithm based on our error estimator are discussed in the numerical section. For the problems considered here, both, the efficiency of the estimator and the quality of the generated meshes are satisfactory.
100 citations
TL;DR: It is asserted that the distance of the piecewise constant discrete gradient to any continuous piecewise affine approximation is a reliable upper error bound up to known higher order terms, consistency terms, and a multiplicative constant.
Abstract: The reliability of frequently applied averaging techniques for a posteriori error control has recently been established for a series of finite element methods in the context of second-order partial differential equations. This paper establishes related reliable and efficient a posteriori error estimates for the energy-norm error of an obstacle problem on unstructured grids as a model example for variational inequalities. The surprising main result asserts that the distance of the piecewise constant discrete gradient to any continuous piecewise affine approximation is a reliable upper error bound up to known higher order terms, consistency terms, and a multiplicative constant.
TL;DR: The coupling preserves in a weak sense the continuity of the solution at the interface without imposing the overall conservativity of the coupled model.
Abstract: We study the theoretical and numerical coupling of two general hyperbolic conservation laws. The coupling preserves in a weak sense the continuity of the solution at the interface without imposing the overall conservativity of the coupled model. In order to analyze the convergence of the coupled numerical scheme, we first revisit the approximation of the boundary value problems. We then prove the convergence and characterize the limit solution of the coupled schemes in a few simple but significative coupling situations. The general coupling problem is analyzed for Riemann initial data and illustrated by numerical simulations. Resume. Nous nous interessons a une nouvelle forme de couplage de deux systemes hyperboliques de lois de conservation. Ce couplage assure de facon faible la continuite de la solution a l’interface sans imposer la conservativite du modele couple. Pour etudier la convergence du schema d’approximation numerique, nous commencons par reprendre les resultats concernant l’approximation du probleme aux limites. Nous demontrons ensuite la convergence du schema couple dans un certain nombre de cas interessants. Le cas general du couplage est etudie et illustre numeriquement pour une donnee initiale de Riemann.
TL;DR: In this paper, a priori and a posteriori estimates for nodal interpolation and orthogonal projection in L2(ΓD) onto the trace space are presented.
Abstract: The numerical solution of elliptic boundary value problems with finite element methods requires the approximation of given Dirichlet data uD by functions uD,h in the trace space of a finite element space on ΓD. In this paper, quantitative a priori and a posteriori estimates are presented for two choices of uD,h, namely the nodal interpolation and the orthogonal projection in L2(ΓD) onto the trace space. Two corresponding extension operators allow for an estimate of the boundary data approximation in global H1 and L2 a priori and a posteriori error estimates. The results imply that the orthogonal projection leads to better estimates in the sense that the influence of the approximation error on the estimates is of higher order than for the nodal interpolation.
TL;DR: It is shown that these wavefunctions possess certain square integrable mixed weak derivatives of order up to N+1 with N the number of electrons, across the singularities of the interaction potentials.
Abstract: The electronic Schrodinger equation describes the motion of electrons under Coulomb interaction forces in the field of clamped nuclei and forms the basis of quantum chemistry. The present article is devoted to the regularity properties of the corresponding wavefunctions that are compatible with the Pauli principle. It is shown that these wavefunctions possess certain square integrable mixed weak derivatives of order up to N+1 with N the number of electrons, across the singularities of the interaction potentials. The result is of particular importance for the analysis of approximation methods that are based on the idea of sparse grids or hyperbolic cross spaces. It indicates that such schemes could represent a promising alternative to current methods for the solution of the electronic Schrodinger equation and that it may even be possible to reduce the computational complexity of an N-electron problem to that of a one-electron problem.
TL;DR: The asymptotic expansions of eigenvalues and eigenfunctions for this kind of problem are obtained, and the multiscale finite element algorithms and numerical results are proposed.
Abstract: In this paper, we study the spectral properties of Dirichlet problems for second order elliptic equation with rapidly oscillating coefficients in a perforated domain. The asymptotic expansions of eigenvalues and eigenfunctions for this kind of problem are obtained, and the multiscale finite element algorithms and numerical results are proposed.
TL;DR: A consistent estimator is proposed, based on a proper correction of the ordinary least squares estimator, which is explicitly given in terms of the true value of the noise variance.
Abstract: A parameter estimation problem for ellipsoid fitting in the presence of measurement errors is considered. The ordinary least squares estimator is inconsistent, and due to the nonlinearity of the model, the orthogonal regression estimator is inconsistent as well, i.e., these estimators do not converge to the true value of the parameters, as the sample size tends to infinity. A consistent estimator is proposed, based on a proper correction of the ordinary least squares estimator. The correction is explicitly given in terms of the true value of the noise variance.
TL;DR: The convergence of a semi-discretized version of a numerical scheme for a stochastic nonlinear Schrödinger equation is studied and it is proved that the discrete solutions in various topologies are convergence.
Abstract: We study the convergence of a semi-discretized version of a numerical scheme for a stochastic nonlinear Schrodinger equation. The nonlinear term is a power law and the noise is multiplicative with a Stratonovich product. Our scheme is implicit in the deterministic part of the equation as is usual for conservative equations. We also use an implicit discretization of the noise which is better suited to Stratonovich products. We consider a subcritical nonlinearity so that the energy can be used to obtain an a priori estimate. However, in the semi discrete case, no Ito formula is available and we have to use a discrete form of this tool. Also, in the course of the proof we need to introduce a cut-off of the diffusion coefficient, which allows to treat the nonlinearity. Then, we prove convergence by a compactness argument. Due to the presence of noise and to the implicit discretization of the noise, this is rather complicated and technical. We finally obtain convergence of the discrete solutions in various topologies.
TL;DR: It is shown that the use of efficient preconditioners does not require to change the energy norm used by the stopping criterion and the results of several numerical tests are presented that experimentally validate the effectiveness of this stopping criterion.
Abstract: The Conjugate Gradient method has always been successfully used in solving the symmetric and positive definite systems obtained by the finite element approximation of self-adjoint elliptic partial differential equations. Taking into account recent results [13, 19, 20, 22] which make it possible to approximate the energy norm of the error during the conjugate gradient iterative process, we adapt the stopping criterion introduced in [3]. Moreover, we show that the use of efficient preconditioners does not require to change the energy norm used by the stopping criterion. Finally, we present the results of several numerical tests that experimentally validate the effectiveness of our stopping criterion.
TL;DR: A fully discrete finite element approximation of the nonlinear cross-diffusion population model of the ith species model is considered and it is proved convergence in space dimensions d≤3.
Abstract: We consider a fully discrete finite element approximation of the nonlinear cross-diffusion population model: Find ui, the population of the ith species, i=1 and 2, such that * where j≠i and gi(u1,u2):=(μi−γii ui−γij uj) ui. In the above, the given data is as follows: v is an environmental potential, ci ∈ ℝ, ai ∈ ℝ are diffusion coefficients, bi ∈ ℝ are transport coefficients, μi ∈ ℝ are the intrinsic growth rates, and γii ∈ ℝ are intra-specific, whereas γij, i≠j, ∈ ℝ are interspecific competition coefficients. In addition to showing well-posedness of our approximation, we prove convergence in space dimensions d≤3. Finally some numerical experiments in one space dimension are presented.
TL;DR: The discretization of the corresponding problems and its influence on the splitting error in terms of the previously conducted numerical analysis of operator splitting techniques for nonlinear reaction-diffusion systems with an entropic structure in the presence of fast scales in the reaction term is investigated.
Abstract: In this paper, we perform the numerical analysis of operator splitting techniques for nonlinear reaction-diffusion systems with an entropic structure in the presence of fast scales in the reaction term. We consider both linear diagonal and quasi-linear non-diagonal diffusion; the entropic structure implies the well-posedness and stability of the system as well as a Tikhonov normal form for the nonlinear reaction term [23]. It allows to perform a singular perturbation analysis and to obtain a reduced and well-posed system of equations on a partial equilibrium manifold as well as an asymptotic expansion of the solution. We then conduct an error analysis in this particular framework where the time scale associated to the fast part of the reaction term is much shorter that the splitting time step Δt thus leading to the failure of the usual splitting analysis techniques. We define the conditions on diffusion and reaction for the order of the local error associated with the time splitting to be reduced or to be preserved in the presence of fast scales. All the results obtained theoretically on local error estimates are then illustrated on a numerical test case where the global error clearly reproduces the scenarios foreseen at the local level. We finally investigate the discretization of the corresponding problems and its influence on the splitting error in terms of the previously conducted numerical analysis.
TL;DR: This work presents a fast and numerically stable algorithm for computing the eigendecomposition of a symmetric block diagonal plus semiseparable matrix that is significantly faster than the standard method which treats the given matrix as a general symmetric dense matrix.
Abstract: We present a fast and numerically stable algorithm for computing the eigendecomposition of a symmetric block diagonal plus semiseparable matrix. We report numerical experiments that indicate that our algorithm is significantly faster than the standard method which treats the given matrix as a general symmetric dense matrix.
TL;DR: An explicit convergent finite difference scheme for motion of level sets by mean curvature is presented and numerical results suggest that despite the low accuracy, acceptable results are achieved for small stencil sizes.
Abstract: An explicit convergent finite difference scheme for motion of level sets by mean curvature is presented. The scheme is defined on a cartesian grid, using neighbors arranged approximately in a circle. The accuracy of the scheme, which depends on the radius of the circle, dx, and on the angular resolution, dθ, is formally O(dx2+dθ). The scheme is explicit and nonlinear: the update involves computing the median of the values at the neighboring grid points. Numerical results suggest that despite the low accuracy, acceptable results are achieved for small stencil sizes. A numerical example is presented which shows that the centered difference scheme is non-convergent.
TL;DR: The relaxation term is reformulate as a nonconservative product regularized by steady-state curves while keeping the velocity variable continuous to ensure realizability and efficiency in efficient numerical simulation of the radiative transfer equations.
Abstract: We are concerned with efficient numerical simulation of the radiative transfer equations. To this end, we follow the Well-Balanced approach's canvas and reformulate the relaxation term as a nonconservative product regularized by steady-state curves while keeping the velocity variable continuous. These steady-state equations are of Fredholm type. The resulting upwind schemes are proved to be stable under a reasonable parabolic CFL condition of the type Δt≤O(Δx2) among other desirable properties. Some numerical results demonstrate the realizability and the efficiency of this process.
TL;DR: A new algorithm for the fast evaluation of linear combinations of radial functions based on the recently developed fast Fourier transform at nonequispaced knots is developed, which proves error estimates to obtain clues about the choice of the involved parameters.
Abstract: We develop a new algorithm for the fast evaluation of linear combinations of radial functions * based on the recently developed fast Fourier transform at nonequispaced knots. For smooth kernels, e.g. the Gaussian, our algorithm requires * arithmetic operations. In case of singular kernels an additional regularization procedure must be incorporated and the algorithm has the arithmetic complexity * if either the points yj or the points xk are “reasonably uniformly distributed”. We prove error estimates to obtain clues about the choice of the involved parameters and present numerical examples for various singular and smooth kernels in two dimensions.
TL;DR: This paper completely explains an excellent long-time behaviour of planetary motions with special linear multistep methods by studying the modified equation of these methods and by analyzing the remarkably stable propagation of parasitic solution components.
Abstract: For computations of planetary motions with special linear multistep methods an excellent long-time behaviour is reported in the literature, without a theoretical explanation. Neither the total energy nor the angular momentum exhibit secular error terms. In this paper we completely explain this behaviour by studying the modified equation of these methods and by analyzing the remarkably stable propagation of parasitic solution components.
TL;DR: It is proved that a standard second order finite difference uniform space discretization of the semilinear wave equation with periodic boundary conditions, analytic nonlinearity, and analytic initial data conserves momentum up to an error which is exponentially small in the stepsize.
Abstract: We prove that a standard second order finite difference uniform space discretization of the semilinear wave equation with periodic boundary conditions, analytic nonlinearity, and analytic initial data conserves momentum up to an error which is exponentially small in the stepsize. Our estimates are valid for as long as the trajectories of the full semilinear wave equation remain real analytic. The method of proof is that of backward error analysis, whereby we construct a modified equation which is itself Lagrangian and translation invariant, and therefore also conserves momentum. This modified equation interpolates the semidiscrete system for all time, and we prove that it remains exponentially close to the trigonometric interpolation of the semidiscrete system. These properties directly imply approximate momentum conservation for the semidiscrete system. We also consider discretizations that are not variational as well as discretizations on non-uniform grids. Through numerical example as well as arguments from geometric mechanics and perturbation theory we show that such methods generically do not approximately preserve momentum.
TL;DR: In this article, the hypersingular integral equation of the first kind equivalently describes screen and Neumann problems on an open surface piece and a computable upper error bound for its Galerkin approximation is established.
Abstract: The hypersingular integral equation of the first kind equivalently describes screen and Neumann problems on an open surface piece. The paper establishes a computable upper error bound for its Galerkin approximation and so motivates adaptive mesh refining algorithms. Numerical experiments for triangular elements on a screen provide empirical evidence of the superiority of adapted over uniform mesh-refining. The numerical realisation requires the evaluation of the hypersingular integral operator at a source point; this and other details on the algorithm are included.
TL;DR: It is proved that the inf-sup constant β satisfies β≤1 and that, if σ=1+ρν−1, the iteration converges linearly with a contraction factor β2ασ-1(2σ-α) provided 0<α<2σ, which yields the optimal value α=σ regardless of β.
Abstract: We consider the Uzawa method to solve the stationary Stokes equations discretized with stable finite elements. An iteration step consists of a velocity update un+1 involving the (augmented Lagrangian) operator −νΔ−ρ∇÷ with ρ≥0, followed by the pressure update pn+1=pn−ανdiv un+1, the so-called Richardson update. We prove that the inf-sup constant β satisfies β≤1 and that, if σ=1+ρν−1, the iteration converges linearly with a contraction factor β2ασ-1(2σ-α) provided 0<α<2σ. This yields the optimal value α=σ regardless of β.
TL;DR: The convergence of a fourth order finite difference method for the 2-D unsteady, viscous incompressible Boussinesq equations, based on the vorticity-stream function formulation, is established in this article.
Abstract: The convergence of a fourth order finite difference method for the 2-D unsteady, viscous incompressible Boussinesq equations, based on the vorticity-stream function formulation, is established in this article. A compact fourth order scheme is used to discretize the momentum equation, and long-stencil fourth order operators are applied to discretize the temperature transport equation. A local vorticity boundary condition is used to enforce the no-slip boundary condition for the velocity. One-sided extrapolation is used near the boundary, dependent on the type of boundary condition for the temperature, to prescribe the temperature at ‘‘ghost’’ points lying outside of the computational domain. Theoretical results of the stability and accuracy of the method are also provided. In numerical experiments the method has been shown to be capable of producing highly resolved solutions at a reasonable computational cost.
TL;DR: A mixed finite element discretization is applied to Richards’ equation, a nonlinear, possibly degenerate parabolic partial differential equation modeling water flow through porous medium, which includes both variably and fully saturated flow regime.
Abstract: A mixed finite element discretization is applied to Richards' equation, a nonlinear, possibly degenerate parabolic partial differential equation modeling water flow through porous medium. The equation is considered in its pressure formulation and includes both variably and fully saturated flow regime. Characteristic for such problems is the lack in regularity of the solution. To handle this we use a time-integrated scheme. We analyze the scheme and present error estimates showing its convergence.