A multigrid method for the Cahn-Hilliard equation with obstacle potential

We present a multigrid finite element method for the deep quench obstacle Cahn-Hilliard equation. The non-smooth nature of this highly nonlinear fourth order partial differential equation make this problem particularly challenging. The method exhibits mesh-independent convergence properties in practice for arbitrary time step sizes. In addition, numerical evidence shows that this behaviour extends to small values of the interfacial parameter γ. Several numerical examples are given, including comparisons with existing alternative solution methods for the Cahn-Hilliard equation.     

Verification for existence of solutions for some generalized obstacle problems

We describe some numerical methods to automatically prove the existence of solutions for some generalized obstacle problems. In this paper, our goal is to establish a new procedure for numerical verification of generalized obstacle problems.     

Monotone convergence of finite element approximations of obstacle problems

The purpose of the work is to study the monotone convergence of numerical solutions of obstacle problems under mesh refinement when the obstacle is convex. We prove monotone convergence of piecewise linear finite element approximations for one-dimensional obstacle problems. We demonstrate by giving a example that such monotone convergence will not hold in the two-dimensional case.     

Optimal shape design for fluid flow using topological perturbation technique

This paper is concerned with an optimal shape design problem in fluid mechanics. The fluid flow is governed by the Stokes equations. The theoretical analysis and the numerical simulation are discussed in two and three-dimensional cases. The proposed approach is based on a sensitivity analysis of a design function with respect to the insertion of a small obstacle in the fluid flow domain. An asymptotic expansion is derived for a large class of cost functions using small topological perturbation technique. A fast and accurate numerical algorithm is proposed. The efficiency of the method is illustrated by some numerical examples.     

A priori error estimates for the finite element approximation of an obstacle problem

The purpose of this paper is to measure, with explicit constants as small as possible, a priori error bounds for approximation by picewise polynomials. These constants play an important role in the numerical verification method of solutions for obstacle problems by using finite element methods.     

Hierarchical error estimates for the energy functional in obstacle problems

We present a hierarchical a posteriori error analysis for the minimum value of the energy functional in symmetric obstacle problems. The main result is that the error in the energy minimum is, up to oscillation terms, equivalent to an appropriate hierarchical estimator. The proof does not invoke any saturation assumption. We even show that small oscillation implies a related saturation assumption. In addition, we prove efficiency and reliability of an a posteriori estimate of the discretization error and thereby cast some light on the theoretical understanding of previous hierarchical estimators. Finally, we illustrate our theoretical results by numerical computations.     

Numerical Approximation of a Unilateral Obstacle Problem

We consider a reformulation of the unilateral obstacle problem presented by the authors (Addou and Mermri in Math-Rech. Appl. 2:59–69, 2000). This reformulation introduces a continuous function, whose subdifferential characterizes the noncontact domain. Our goal in this paper is to give a numerical approximation of the solution of the reformulated problem. We consider discretization of the problem based on finite element method. Then we prove the convergence of the approximate solution to the exact one. Some numerical tests on one-dimensional obstacle problem are provided.     

A dual interpolation boundary face method with Hermite-type approximation for potential problems

This paper presents the dual interpolation boundary face method combined with a Hermite-type moving-least-squares approximation for solving complex two-dimensional potential problems. Compared to the standard algorithms, this combined method is better suited for structures with small feature sizes such as short edges and small chamfers. The interpolation functions, if constructed in cyclic coordinates, making it difficult to apply this new method to deal with complex structures with small feature sizes in which only one source point is assigned. The Hermite-type approximation formulated in Cartesian coordinates is able to completely overcome this obstacle by searching for source points on adjacent edges. Additionally, an improved and incomplete quadratic polynomial basis is presented to obtain an accurate algorithm for the Hermite-type approximation. We use several numerical examples to demonstrate the high accuracy and efficiency of the proposed method for solving various engineering structures with small feature sizes.     

Furtivity and Masking Problems in Time-Dependent Electromagnetic Obstacle Scattering

In this paper, we consider furtivity and masking problems in time-dependent three-dimensional electromagnetic obstacle scattering. That is, we propose a criterion based on a merit function to minimize or to mask the electromagnetic field scattered by a bounded obstacle when hit by an incoming electromagnetic field and, with respect to this criterion, we drive the optimal strategy. These problems are natural generalizations to the context of electromagnetic scattering of the furtivity problem in time-dependent acoustic obstacle scattering presented in Ref. 1. We propose mathematical models of the furtivity and masking time-dependent three-dimensional electromagnetic scattering problems that consist in optimal control problems for systems of partial differential equations derived from the Maxwell equations. These control problems are approached using the Pontryagin maximum principle. We formulate the first-order optimality conditions for the control problems considered as exterior problems defined outside the obstacle for systems of partial differential equations. Moreover, the first-order optimality conditions derived are solved numerically with a highly parallelizable numerical method based on a perturbative series of the type considered in Refs. 2–3. Finally, we assess and validate the mathematical models and the numerical method proposed analyzing the numerical results obtained with a parallel implementation of the numerical method in several experiments on test problems. Impressive speedup factors are obtained executing the algorithms on a parallel machine when the number of processors used in the computation ranges between 1 and 100. Some virtual reality applications and some animations relative to the numerical experiments can be found in the website http://www.econ.unian.it/recchioni/w10/.     

A Dynamical Method for Solving the Obstacle Problem

In this paper, we consider the unilateral obstacle problem, trying to find the numerical solution and coincidence set. We construct an equivalent format of the original problem and propose a method with a second-order in time dissipative system for solving the equivalent format. Several numerical examples are given to illustrate the effectiveness and stability of the proposed algorithm. Convergence speed comparisons with existent numerical algorithm are also provided and our algorithm is fast.     

Parametric Cubic Spline Approach to the Solution of a System of Second-Order Boundary-Value Problems

We use parametric cubic spline functions to develop a numerical method for computing approximations to the solution of a system of second-order boundary-value problems associated with obstacle, unilateral, and contact problems. We show that the present method gives approximations which are better than those produced by other collocation, finite-difference, and spline methods. A numerical example is given to illustrate the applicability and efficiency of the new method.     

Finite-Difference Method for a System of Third-Order Boundary-Value Problems

In this paper, we develop a two-stage numerical method for computing the approximate solutions of third-order boundary-value problems associated with odd-order obstacle problems. We show that the present method is of order two. A numerical example is presented to illustrate the applicability of the new method. A comparison is also given with previously known results.     

A Priori Error Estimates for Obstacle Optimal Control Problem,Where the Obstacle Is the Control Itself

In this paper we deal with the convergence analysis of the finite element method for an elliptic penalized unilateral obstacle optimal control problem where the control and the obstacle coincide. Error estimates are established for both state and control variables. We apply a fixed point type iteration method to solve the discretized problem.To corroborate our error estimations and the efficiency of our algorithms, the convergence results and numerical experiments are illustrated by concrete examples.     

Sextic spline method for the solution of a system of obstacle problems

We use sextic spline function to develop numerical method for the solution of system of second-order boundary-value problems associated with obstacle, unilateral, and contact problems. We show that the approximate solutions obtained by the present method are better than those produced by other collocation, finite difference and spline methods. A numerical example is given to illustrate practical usefulness of our method.     

A Probabilistic-Numerical Approximation for an Obstacle Problem Arising in Game Theory

We investigate a two-player zero-sum stochastic differential game in which one of the players has more information on the game than his opponent. We show how to construct numerical schemes for the value function of this game, which is given by the solution of a quasilinear partial differential equation with obstacle.     

Optimal Control of the Obstacle Problem in a Perforated Domain

We study the problem of optimally controlling the solution of the obstacle problem in a domain perforated by small periodically distributed holes. The solution is controlled by the choice of a perforated obstacle which is to be chosen in such a fashion that the solution is close to a given profile and the obstacle is not too irregular. We prove existence, uniqueness and stability of an optimal obstacle and derive necessary and sufficient conditions for optimality. When the number of holes increase indefinitely we determine the limit of the sequence of optimal obstacles and solutions. This limit depends strongly on the rate at which the size of the holes shrink.     

Numerical solutions for system of second order boundary value problems

We investigate some numerical methods for computing approximate solutions of a system of second order boundary value problems associated with obstacle, unilateral and contact problems. We show that cubic spline method gives approximations which are better than that computed by higher order spline and finite difference techniques.     

Recovering the Dirichlet-to-Neumann map in inverse scattering problems using integral equation methods

Consider the reconstruction of Dirichlet-to-Neumann map(D-to-N map) from the far-field patterns of the scattered waves in inverse scattering problems, which is the first step in detecting the obstacle boundary by the probe method using far-field measurements corresponding to all incident plane waves. In principle, this problem can be reduced to solving an integral equation of the second kind with the kernels involving the derivatives of the scattered waves for point sources. Based on the mixed reciprocity principle, we propose two simple and feasible numerical schemes for reconstructing D-to-N map. Compared with the well-known obstacle boundary recovering schemes using the simulation of D-to-N map directly, the proposed schemes give the possible ways to realizing the probe methods using practical far-field data, with the advantage of no numerical differentiation for scattered wave in their implementations. We present some numerical examples for the D-to-N map, showing the validity and stability of our schemes.     

Laminar magnetohydrodynamic duct flow in the presence of a magnetic dipole

We study magnetohydrodynamic flow of a liquid metal in a straight duct. The magnetic field is produced by an exterior magnetic dipole. This basic configuration is of fundamental interest for Lorentz force velocimetry (LFV), where the Lorentz force opposing the relative motion of conducting medium and magnetic field is measured to determine the flow velocity. The Lorentz force acts in equal strength but opposite direction on the flow as well as on the dipole. We are interested in the dependence of the velocity on the flow rate and on strength of the magnetic field as well as on geometric parameters such as distance and position of the dipole relative to the duct. To this end, we perform numerical simulations with an accurate finite-difference method in the limit of small magnetic Reynolds number, whereby the induced magnetic field is assumed to be small compared with the external applied field. The hydrodynamic Reynolds number is also assumed to be small so that the flow remains laminar. The simulations allow us to quantify the magnetic obstacle effect as a potential complication for local flow measurement with LFV. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Simple Semi-Implicit Scheme for Partial Differential Equations with Obstacle Constraints

We develop a simple and efficient numerical scheme to solve a class of obstacle problems encountered in various applications. Mathematically, obstacle problems are usually formulated using nonlinear partial differential equations (PDE). To construct a computationally efficient scheme, we introduce a time derivative term and convert the PDE into a time-dependent problem. But due to its nonlinearity, the time step is in general chosen to satisfy a very restrictive stability condition. To relax such a time step constraint when solving a time dependent evolution equation, we decompose the nonlinear obstacle constraint in the PDE into a linear part and a nonlinear part and apply the semi-implicit technique. We take the linear part implicitly while treating the nonlinear part explicitly. Our method can be easily applied to solve the fractional obstacle problem and min curvature flow problem. The article will analyze the convergence of our proposed algorithm. Numerical experiments are given to demonstrate the efficiency of our algorithm.