Existence of a Classical Solution for a Multidimensional Filtration Problem

We prove existence, in a small time interval, of a classical solution having a regular free boundary, (i.e. the level set {u=0}) of a multidimensional filtration problem. The key point to get the regularity of the level set {u=0} is the proof of an a priori estimate for the L_∞-norm of u_t. © 1997 by B.G. Teubner Stuttgart-John Wiley & Sons, Ltd.     

Higher order of fully discreate solution for parabolic problem inL ∞

In this work, we approximate the solution of initial boundary value problem using a Galerkin-finite element method for the spatial discretization, and Implicit Runge-Kutta methods for the time stepping. To deal with the nonlinear termf(x, t, u), we introduce the well-known extrapolation sheme which was used widely to prove the convergence inL ₂-norm. We present computational results showing that the optimal order of convergence arising underL ₂-norm will be preserved inL _∞-norm.     

The regulator problem for parabolic equations with Dirichlet boundary control

This paper considers the regulator problem for a parabolic equation (generally unstable), defined on an open, bounded domain , with control functionu acting in the Dirichlet boundary condition: minimize the quadratic functional which penalizes theL ₂(0, ; L₂())-norm of the solutiony and theL ₂(0, ; L₂())-norm of the Dirichlet controlu. The paper is divided in two parts. Part I derives, in a constructive way, the algebraic Riccati equation satisfied by the candidate Riccati operator solution (unique in our case) and, moreover, studies the regularity properties of the optimal pairu ⁰, y⁰. Part II studies a Galerkin approximation of the regulator problem. It shows first the uniform analyticity and the uniform exponential stability of the underlying discrete (approximating) semigroups. Then it establishes the desired convergence properties, in particular, pointwise Riccati operators convergence and, as a final goal, convergence of the original dynamics acted upon by the discrete feedbacks.Research partially supported by the National Science Foundation under Grant DMS-8301668.     

Multiple Degree Reduction and Elevation of Bézier Curves Using Jacobi–Bernstein Basis Transformations

In this article, we find the optimal r times degree reduction of Bézier curves with respect to the Jacobi-weighted L ₂-norm on the interval [0, 1]. This method describes a simple and efficient algorithm based on matrix computations. Also, our method includes many previous results for the best approximation with L ₁, L ₂, and L _∞-norms. We give some examples and figures to demonstrate these methods.     

Monte-Carlo approximations for 2d Navier-Stokes equations with measure initial data

We are interested in proving Monte-Carlo approximations for 2d Navier-Stokes equations with initial data u ₀ belonging to the Lorentz space L ^2,∞ and such that curl u ₀ is a finite measure. Giga, Miyakawaand Osada [7] proved that a solution u exists and that u=K* curl u, where K is the Biot-Savartkernel and v = curl u is solution of a nonlinear equation in dimension one, called the vortex equation. In this paper, we approximate a solution v of this vortex equationby a stochastic interacting particlesystem and deduce a Monte-Carlo approximation for a solution of the Navier-Stokesequation. That gives in this case a pathwise proofof the vortex algorithm introducedby Chorin and consequently generalizes the works ofMarchioro-Pulvirenti [12] and Méléardv [15] obtained in the case of a vortex equation with bounded density initial data. Received: 6 October 1999 / Revised version: 15 September 2000 / Published online: 9 October 2001     

Correctors for some asymptotic problems

In the theory of anisotropic singular perturbation boundary value problems, the solution u _ɛ does not converge, in the H ¹-norm on the whole domain, towards some u ₀. In this paper we construct correctors to have good approximations of u _ɛ in the H ¹-norm on the whole domain. Since the anisotropic singular perturbation problems can be connected to the study of the asymptotic behaviour of problems defined in cylindrical domains becoming unbounded in some directions, we transpose our results for such problems.     

Superconvergence of tricubic block finite elements

In this paper, we first introduce interpolation operator of projection type in three dimen- sions, from which we derive weak estimates for tricubic block finite elements. Then using the estimate for the W 2, 1-seminorm of the discrete derivative Green’s function and the weak estimates, we show that the tricubic block finite element solution uh and the tricubic interpolant of projection type Πh3u have superclose gradient in the pointwise sense of the L∞-norm. Finally, this supercloseness is applied to superc...     

On the approximate calculation of time-minimal distributed controls of vibrations by finite element approximations

This paper is concerned with distributed null-control of vibrations governed by an abstract wave equation. Based on a method for the exact computation of minimumL ²-norm controls for given time intervals and time-minimal controls which are bounded with respect to theL ²-norm, an approximation method is developed which is based on Galerkin's method ana convergence results are derived.This paper is based on U. Lamp's doctoral dissertation and was supported by the Deutsche Forschungsgemeinschaft.     

An optimal order error estimate for an upwind discretization of the Navier—Stokes equations

We analyze a finite-element approximation of the stationary incompressible Navier–Stokes equations in primitive variables. This approximation is based on the nonconforming P₁/P₀ element pair of Crouzeix/Raviart and a special upwind discretization of the convective term. An optimal error estimate in a discrete H¹-norm for the velocity and in the L²-norm for the pressure is proved. Some numerical results are presented. © 1996 John Wiley & Sons, Inc.     

L1-regularisation for ill-posed problems in variational data assimilation

We consider four-dimensional variational data assimilation (4DVar) and show that it can be interpreted as Tikhonov or L₂-regularisation, a widely used method for solving ill-posed inverse problems. It is known from image restoration and geophysical problems that an alternative regularisation, namely L₁-norm regularisation, recovers sharp edges better than L₂-norm regularisation. We apply this idea to 4DVar for problems where shocks and model error are present and give two examples which show that L₁-norm regularisation performs much better than the standard L₂-norm regularisation in 4DVar. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A robust semi-explicit difference scheme for the Kuramoto–Tsuzuki equation

In this paper, we propose a robust semi-explicit difference scheme for solving the Kuramoto–Tsuzuki equation with homogeneous boundary conditions. Because the prior estimate in L_∞-norm of the numerical solutions is very hard to obtain directly, the proofs of convergence and stability are difficult for the difference scheme. In this paper, we first prove the second-order convergence in L₂-norm of the difference scheme by an induction argument, then obtain the estimate in L_∞-norm of the numerical solutions. Furthermore, based on the estimate in L_∞-norm, we prove that the scheme is also convergent with second order in L_∞-norm. Numerical examples verify the correction of the theoretical analysis.     

Application of Rothe’s Method to a Parabolic Inverse Problem with Nonlocal Boundary Condition

We consider an inverse problem for the determination of a purely time-dependent source in a semilinear parabolic equation with a nonlocal boundary condition. An approximation scheme for the solution together with the well-posedness of the problem with the initial value u₀ ∈ H¹(Ω) is presented by means of the Rothe time-discretization method. Further approximation scheme via Rothe’s method is constructed for the problem when u₀ ∈ L²(Ω) and the integral kernel in the nonlocal boundary condition is symmetric.

     

Solution of two-dimensional boundary value problems corresponding to initial-boundary value problems of diffusion on a right cylinder

We consider boundary value problems for the differential equations Δ₂ u + B u = 0 with operator coefficients B corresponding to initial-boundary value problems for the diffusion equation Δ₃ u − pu = ∂ _t u (p > 0) on a right cylinder with inhomogeneous boundary conditions on the lateral surface of the cylinder with zero boundary conditions on the bases of the cylinder and with zero initial condition. For their solution, we derive specific boundary integral equations in which the space integration is performed only over the lateral surface of the cylinder and the kernels are expressed via the fundamental solution of the two-dimensional heat equation and the Green function of corresponding one-dimensional initial-boundary value problems of diffusion. We prove uniqueness theorems and obtain sufficient existence conditions for such solutions in the class of functions with continuous L ₂-norm.     

On the global weak solutions for a modified two‐component Camassa‐Holm equation

In this paper, we investigate the existence of global weak solutions to the Cauchy problem of a modified two‐component Camassa‐Holm equation with the initial data satisfying lim_{x → ±∞}u₀(x) = u_±. By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution for the system under the assumption u_? ≤ u₊. The global weak solution is obtained as a limit of approximation solutions. The key elements in our analysis are the Helly theorem and the estimation of energy for approximation solutions in $H^1(\mathbb {R})\times H^1(\mathbb {R})In this paper, we investigate the existence of global weak solutions to the Cauchy problem of a modified two‐component Camassa‐Holm equation with the initial data satisfying lim_{x → ±∞}u₀(x) = u_±. By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution for the system under the assumption u_? ≤ u₊. The global weak solution is obtained as a limit of approximation solutions. The key elements in our analysis are the Helly theorem and the estimation of energy for approximation solutions in $H^1(\mathbb {R})\times H^1(\mathbb {R})$ and some a priori estimates on the first‐order derivatives of approximation solutions.     

Estimates for the L 2-Projection onto Continuous Finite Element Spaces in a Weighted L p -Norm

We consider the orthogonal L₂-projection P onto continuous finite element spaces. We prove estimates for P in a weighted L_p-norm and use these to prove corresponding approximation properties. AMS subject classification (2000) 65N60, 35K85     

Exact L 2-norm plane separation

We consider the problem of separating two sets of points in an n-dimensional real space with a (hyper)plane that minimizes the sum of L _p -norm distances to the plane of points lying on the wrong side of it. Despite recent progress, practical techniques for the exact solution of cases other than the L ₁ and L _∞-norm were unavailable. We propose and implement a new approach, based on non-convex quadratic programming, for the exact solution of the L ₂-norm case. We solve in reasonable computing times artificial problems of up to 20000 points (in 6 dimensions) and 13 dimensions (with 2000 points). We also observe that, for difficult real-life instances from the UCI Repository, computation times are substantially reduced by incorporating heuristic results in the exact solution process. Finally, we compare the classification performance of the planes obtained for the L ₁, L ₂ and L _∞ formulations. It appears that, despite the fact that L ₂ formulation is computationally more expensive, it does not give significantly better results than the L ₁ and L _∞ formulations.     

On Bernstein–Markov-type inequalities for multivariate polynomials in -norm

Bernstein–Markov-type inequalities provide estimates for the norms of derivatives of algebraic and trigonometric polynomials. They play an important role in Approximation Theory since they are widely used for verifying inverse theorems of approximation. In the past decades these inequalities were extended to the multivariate setting, but the main emphasis so far was on the uniform norm. It is considerably harder to derive Bernstein–Markov-type inequalities in the L_q-norm, and it requires introduction of new methods. In this paper we verify certain Bernstein–Markov-type inequalities in L_q-norm on convex and star-like domains. Special attention is given to the question of how the geometry of the domain affects the corresponding estimates.     

Approximations of linear control problems with bang-bang solutions

We analyse the Euler discretization to a class of linear optimal control problems. First we show convergence of order h for the discrete approximation of the adjoint solution and the switching function, where h is the mesh size. Under the additional assumption that the optimal control has bang-bang structure we show that the discrete and the exact controls coincide except on a set of measure O(h). As a consequence, the discrete optimal control approximates the optimal control with order 1 w.r.t. the L ¹-norm and with order 1/2 w.r.t. the L ²-norm. An essential assumption is that the slopes of the switching function at its zeros are bounded away from zero which is in fact an inverse stability condition for these zeros. We also discuss higher order approximation methods based on the approximation of the adjoint solution and the switching function. Several numerical examples underline the results.     

Zur Abhängigkeit der Lösung von ut + (h(u))x = 0 von ihren Anfangswerten

The above equation has some remarkable properties. In general a global solution exists in a weak sense only, and this solution is not reversible in time. Furthermore it is known, that the solutions for different initial values can coincide for all t ? t₀ > 0, and the set of the initial values with this property is convex. Conditions assuring that this set contains only one element are given. This means a weak form of time-reversibility. As a global solution exists only in the weak sense, the classical question concerning dependence of the solution on the initial values needs some modification. This problem is dealt with in suitable L₁-norms. It is shown, that the L₁-norm of the difference of two weak solutions with respect to the space variable does not increase in time.     

A variable coefficient nonlinear Schrödinger equation with a four-dimensional symmetry group and blow-up

A canonical variable coefficient nonlinear Schrödinger equation with a four-dimensional symmetry group containing SL(2,??) group as a subgroup is considered. This typical invariance is then used to transform by a symmetry transformation a known solution that can be derived by truncating its Painlevé expansion and study blow-ups of these solutions in the L _p -norm for p?>?2, L _∞-norm and in the sense of distributions.