Showing papers on "Laplace's equation published in 2004"

Adaptive Finite Element Methods with convergence rates

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.

Fractional heat conduction equation and associated thermal stress

Abstract: A quasi-static uncoupled theory of thermoelasticity based on the heat conduction equation with a time-fractional derivative of order α is proposed. Because the heat conduction equation in the case 1≤α≤2 interpolates the parabolic equation (α = 1) and the wave equation (α = 2), the proposed theory interpolates a classical thermoelasticity and a thermoelasticity without energy dissipation introduced by Green and Naghdi. The Caputo fractional derivative is used. The stresses corresponding to the fundamental solutions of a Cauchy problem for the fractional heat conduction equation are found in one-dimensional and two-dimensional cases.

Transform Methods for Solving Partial Differential Equations

Abstract: THE FUNDAMENTALS Fourier Transforms Laplace Transforms Linear Ordinary Differential Equations Complex Variables Multivalued Functions, Branch Points, Branch Cuts, and Riemann Surfaces Some Examples of Integration which Involve Multivalued Functions Bessel Functions What are Transform Methods? METHODS INVOLVING SINGLE-VALUED LAPLACE TRANSFORMS Inversion of Laplace Transforms by Contour Integration The Heat Equation The Wave Equation Laplace's and Poisson's Equations Papers Using Laplace Transforms to Solve Partial Differential Equations METHODS INVOLVING SINGLE-VALUED FOURIER AND HANKEL TRANSFORMS Inversion of Fourier Transforms by Contour Integration The Wave Equation The Heat Equation Laplace's Equation The Solution of Partial Differential Equations by Hankel Transforms Numerical Inversion of Hankel Transforms Papers Using Fourier Transforms to Solve Partial Differential Equations Papers Using Hankel Transforms to Solve Partial Differential Equations METHODS INVOLVING MULTIVALUED LAPLACE TRANSFORMS Inversion of Laplace Transforms by Contour Integration Numerical Inversion of Laplace Transforms The Wave Equation The Heat Equation Papers Using Laplace Transforms to Solve Partial Differential Equations METHODS INVOLVING MULTIVALUED FOURIER TRANSFORMS Inversion of Fourier Transforms by Contour Integration Numerical Inversion of Fourier Transforms The Solution of Partial Differential Equations by Fourier Transforms Papers Using Fourier Transforms to Solve Partial Differential Equations THE JOINT TRANSFORM METHOD The Wave Equation The Heat and Other Partial Differential Equations Inversion of the Joint Transform by Cagniard's Method The Modification of Cagniard's Method by De Hoop Papers Using the Joint Transform Technique Papers Using the Cagniard Technique Papers Using the Cagniard-De Hoop Technique THE WIENER-HOPF TECHNIQUE The Wiener-Hopf Technique When the Factorization Contains No Branch Points The Wiener-Hopf Technique when the Factorization Contains Branch Points Papers Using the Wiener-Hopf Technique WORKED SOLUTIONS TO SOME OF THE PROBLEMS INDEX

On the Axioms of Scale Space Theory

Abstract: We consider alternative scale space representations beyond the well-established Gaussian case that satisfy all “reasonable” axioms. One of these turns out to be subject to a first order pseudo partial differential equation equivalent to the Laplace equation on the upper half plane l(x, s) ∈ \Bbb Rd × \Bbb R v s > 0r. We investigate this so-called Poisson scale space and show that it is indeed a viable alternative to Gaussian scale space. Poisson and Gaussian scale space are related via a one-parameter class of operationally well-defined intermediate representations generated by a fractional power of (minus) the spatial Laplace operator.

A convergent difference scheme for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions

Abstract: This article considers the problem of building absolutely minimizing Lipschitz extensions to a given function. These extensions can be characterized as being the solution of a degenerate elliptic partial differential equation, the infinity Laplacian, for which there exist unique viscosity solutions. A convergent difference scheme for the infinity Laplacian equation is introduced, which arises by minimizing the discrete Lipschitz constant of the solution at every grid point. Existence and uniqueness of solutions to the scheme is shown directly. Solutions are also shown to satisfy a discrete comparison principle. Solutions are computed using an explicit iterative scheme which is equivalent to solving the parabolic version of the equation.

The evaporation/condensation transition of liquid droplets

Abstract: The condensation of a supersaturated vapor enclosed in a finite system is considered. A phenomenological analysis reveals that the vapor is found to be stable at densities well above coexistence. The system size at which the supersaturated vapor condenses into a droplet is found to be governed by a typical length scale which depends on the coexistence densities, temperature and surface tension. When fluctuations are neglected, the chemical potential is seen to show a discontinuity at an effective spinodal point, where the inhomogeneous state becomes more stable than the homogeneous state. If fluctuations are taken into account, the transition is rounded, but the slope of the chemical potential versus density isotherm develops a discontinuity in the thermodynamic limit. In order to test the theoretical predictions, we perform a simulation study of droplet condensation for a Lennard-Jones fluid and obtain loops in the chemical potential versus density and pressure. By computing probability distributions for the cluster size, chemical potential, and internal energy, we confirm that the effective spinodal point may be identified with the occurrence of a first order phase transition, resulting in the condensation of a droplet. An accurate equation of state is employed in order to estimate the droplet size and the coexisting vapor density and good quantitative agreement with the simulation data is obtained. The results highlight the need of an accurate equation of state data for the Laplace equation to have predictive power.

An oscillating bubble near an elastic material

Abstract: A method is presented to describe the behavior of an oscillating bubble in a fluid near a second elastic (biological) fluid The elasticity of the second fluid is modeled through a pressure term at the interface between the two fluids The Laplace equation is assumed to be valid in each of the fluids, and a difference in the respective densities is allowed A relationship between the two velocity potentials just above and below the fluid-fluid interface can be found The boundary integral method is then used to solve for the unknown normal velocities at both the bubble interface and fluid-fluid interface These said normal velocities are subsequently utilized to update the position of the interface(s) for the next time step For bubbles oscillating near a second nonelastic fluid, the bubbles can develop a jet towards or away from the fluid-fluid interface (depending on the distance of the bubble from the fluid-fluid interface and the density ratios of the two fluids) This behavior can be greatly modified

Asymptotic Approximation of the Solution of the Laplace Equation in a Domain with Highly Oscillating Boundary

Abstract: We study the asymptotic behavior of the solution of the Laplace equation in a domain, a part of whose boundary is highly oscillating. The motivation comes from the study of a longitudinal flow in an infinite horizontal domain bounded at the bottom by a wall and at the top by a rugose wall. The latter is a plane covered with periodic asperities whose size depends on a small parameter, $ \varepsilon >0.$ The assumption of sharp asperities is made; that is, the height of the asperities is fixed. Using a boundary layer corrector, we derive and analyze a nonoscillating approximation of the solution at order ${\cal O}(\varepsilon^{3/2})$ for the H1 -norm.

Logarithmic stability estimates for a Robin coefficient in two-dimensional Laplace inverse problems

Abstract: We establish some global stability results together with logarithmic estimates in Sobolev norms for the inverse problem of recovering a Robin coefficient on part of the boundary of a smooth 2D domain from overdetermined measurements on the complementary part of a solution to the Laplace equation in the domain, using tools from analytic function theory.

Time discretization of an evolution equation via Laplace transforms

Abstract: Following earlier work by Sheen, Sloan, and Thomee concerning parabolic equations we study the discretization in time of a Volterra type integro-differential equation in which the integral operator is a convolution of a weakly singular function and an elliptic differential operator in space. The time discretization is accomplished by using a modified Laplace transform in time to represent the solution as an integral along a smooth curve extending into the left half of the complex plane, which is then evaluated by quadrature. This reduces the problem to a finite set of elliptic equations with complex coefficients, which may be solved in parallel. Stability and error bounds of high order are derived for two different choices of the quadrature rule. The method is combined with finite-element discretization in the spatial variables.

Fractional Laplace model for hydraulic conductivity

Abstract: [1] Based on an examination of K data from four different sites, a new stochastic fractal model, fractional Laplace motion, is proposed. This model is based on the assumption of spatially stationary ln(K) increments governed by the Laplace PDF, with the increments named fractional Laplace noise. Similar behavior has been reported for other increment processes (often called fluctuations) in the fields of finance and turbulence. The Laplace PDF serves as the basis for a stochastic fractal as a result of the geometric central limit theorem. All Laplace processes reduce to their Gaussian analogs for sufficiently large lags, which may explain the apparent contradiction between large-scale models based on fractional Brownian motion and non-Gaussian behavior on smaller scales.

A laboratory on the four-point probe technique

Abstract: We describe how a classic electrostatics experiment can be modified to be a four-point probe lab experiment. Students use the four-point probe technique to investigate how the measured resistance varies as a function of the position of the electrodes with respect to the edge of the sample. By using elementary electromagnetism concepts such as the superposition principle, the continuity equation, the relation between electric field and electric potential, and Ohm’s law, a simple model is derived to describe the four-point probe technique. Although the lab introduces the students to the ideas behind the Laplace equation and the methods of images, advanced mathematics is avoided so that the experiment can be done in trigonometry and algebra based physics courses. In addition, the experiment introduces the students to a standard measurement technique that is widely used in industry and thus provides them with useful hands-on experience.

Direct Evaluation of Hypersingular Galerkin Surface Integrals

Abstract: A direct algorithm for evaluating hypersingular integrals arising in a three-dimensional Galerkin boundary integral analysis is presented. The singular integrals are defined as limits to the boundary, and by integrating two of the four dimensions analytically, the coincident integral is shown to be divergent. However, the divergent terms can be explicitly calculated and shown to cancel with corresponding singularities in the adjacent edge integrals. A single analytic integration is employed for the edge and vertex singular integrals. This is sufficient to display the divergent term in the edge-adjacent integral and to show that the vertex integral is finite. By explicitly identifying the divergent quantities, we can compute the hypersingular integral without recourse to Stokes's theorem or the Hadamard finite part. The algorithms are developed in the context of a linear element approximation for the Laplace equation but are expected to be generally applicable. As an example, the algorithms are applied to solve a thermal problem in an exponentially graded material.

Boundary integral equations as applied to an oscillating bubble near a fluid-fluid interface

Abstract: A new method is presented to describe the behaviour of an oscillating bubble near a fluid-fluid interface. Such a situation can be found for example in underwater explosions (near muddy bottoms) or in bubbles generated near two (biological) fluids separated by a membrane. The Laplace equation is assumed to be valid in both fluids. The fluids can have different density ratios. A relationship between the two velocity potentials just above and below the fluid-fluid interface can be used to update the co-ordinates of the new interface at the next time step. The boundary integral method is then used for both fluids. With the resulting equations the normal velocities on the interface and the bubble are obtained. Depending on initial distances of the bubble from the fluid-fluid interface and density ratios, the bubbles can develop jets towards or away from this interface. Gravity can be important for bubbles with larger dimensions.

Floating Breakwater Response to Waves Action Using a Boussinesq Model Coupled with a 2DV Elliptic Solver

Abstract: The hydrodynamic behavior of fixed and heave motion floating breakwaters is studied in the present paper, using a finite-difference, mathematical model based on the Boussinesq type equations. The flow under the floating breakwater is treated separately as confined flow. The pressure field beneath the floating structure is determined by solving implicitly the Laplace equation for the potential Φ of the confined flow using the appropriate boundary conditions. The dynamic equation of heave motion is solved with the consequent adjustments of the continuity equation in the case of a heave motion floating breakwater. Numerical results, concerning the efficiency of fixed and heave motion floating breakwaters, are compared to experimental results satisfactorily. The ability of the numerical model to predict the pressure field beneath the floating structure and the vertical force acting on it is thoroughly examined by making comparisons of the numerical results with large-scale experimental data. The experiments were conducted in the CIEM flume of the Catalonia Univ. of Technology, Barcelona, Spain. The final goal is to study floating breakwaters efficiency in shallow and intermediate waters.

Lattice Boltzmann Study on the Contact Angle and Contact Line Dynamics of Liquid−Vapor Interfaces

Abstract: The moving contact line problem of liquid-vapor interfaces was studied using a mean-field free-energy lattice Boltzmann method recently proposed [Phys. Rev. E 2004, 69, 032602]. We have examined the static and dynamic interfacial behaviors by means of the bubble and capillary wave tests and found that both the Laplace equation of capillarity and the dispersion relation were satisfied. Dynamic contact angles followed the general trend of contact line velocity observed experimentally and can be described by Blake's theory. The velocity fields near the interface were also obtained and are in good agreement with fluid mechanics and molecular dynamics studies. Our simulations demonstrated that incorporating interfacial effects into the lattice Boltzmann model can be a valuable and powerful alternative in interfacial studies.

Relaxation procedures for an iterative algorithm for solving the Cauchy problem for the Laplace equation

Abstract: In this paper an iterative alternating algorithm for solving the Cauchy problem for the Laplace equation is considered. Three relaxation procedures are developed in order to increase the rate of convergence of the algorithm and selection criteria for the variable relaxation factors are provided. These procedures are analysed and compared both theoretically and numerically. The boundary element method is used in order to implement numerically the iterative algorithm and to show that the ill-posed Cauchy problem is regularized by using an appropriate stopping criterion. The numerical results obtained show that by using the relaxation procedures proposed in this paper, the number of iterations required to achieve convergence may be drastically reduced.

Analysis of Finite Element Approximation of Evolution Problems in Mixed Form

Abstract: This paper deals with the finite element approximation of evolution problems in mixed form. Following [D. Boffi, F. Brezzi, and L. Gastaldi, Math. Comp., 69 (2000), pp. 121--140], we handle separately two types of problems. A model for the first case is the heat equation in mixed form, while the time dependent Stokes problem fits within the second one. For either case, we give sufficient conditions for a good approximation in the natural functional spaces. The results are not obvious in the first situation. In this case, the well-known conditions for the well posedness and convergence of the corresponding steady problem are not sufficient for the good approximation of the time dependent problem. This issue is demonstrated with a numerical (counter-) example and justified analytically.

A general stability criterion for droplets on structured substrates

Abstract: Wetting morphologies on solid substrates, which may be chemically or topographically structured, are studied theoretically by variation of the free energy which contains contributions from the substrate surface, the fluid– fluid interface and the three-phase contact line. The first variation of this free energy leads to two equations—the classical Laplace equation and a generalized contact line equation—which determine stationary wetting morphologies. From the second variation of the free energy we derive a general spectral stability criterion for stationary morphologies. In order to incorporate the constraint that the displaced contact line must lie within the substrate surface, we consider only normal interface displacements but introduce a variation of the domains of parametrization.

Solitary wave and other solutions for nonlinear heat equations

Abstract: A number of explicit solutions for the heat equation with a polynomial non-linearity and for the Fisher equation is presented. An extended class of non-linear heat equations admitting solitary wave solutions is described. The generalization of the Fisher equation is proposed whose solutions propagate with arbitrary ad hoc fixed velocity.

Numerical analysis of the MFS for certain harmonic problems

Abstract: The Method of Fundamental Solutions (MFS) is a boundary-type meshless method for the solution of certain elliptic boundary value problems. In this work, we investigate the properties of the matrices that arise when the MFS is applied to the Dirichlet problem for Laplace's equation in a disk. In particular, we study the behaviour of the eigenvalues of these matrices and the cases in which they vanish. Based on this, we propose a modified efficient numerical algorithm for the solution of the problem which is applicable even in the cases when the MFS matrix might be singular. We prove the convergence of the method for analytic boundary data and perform a stability analysis of the method with respect to the distance of the singularities from the origin and the number of degrees of freedom. Finally, we test the algorithm numerically.

On the Eigenvalues of Finitely Perturbed Laplace Operators in Infinite Cylindrical Domains

Abstract: In this paper, sufficient conditions for the existence of eigenvalues of a finitely perturbed Laplace operator in an infinite cylindrical domain and their asymptotics in the small parameter are given. Similar questions for tubes, i.e., deformed cylinders, are also considered.