Numerical solution of a heat diffusion problem by boundary element methods using the Laplace transform

This paper is concerned with a heat diffusion problem in a half-space which is motivated by the detection of material defects using thermal measurements. This problem is solved by inverting the Laplace transform with respect to time on a contour in the complex plane using an exponentially convergent quadrature rule. This leads to a finite number of time-independent problems, which can be solved in parallel using boundary integral equation methods. We provide a full numerical analysis of this scheme on compact time intervals. Our results are formulated in a way that they can easily be used for other diffusion problems in exterior or interior domains.     

A numerical method for the Benjamin-Ono equation

This paper is concerned with the numerical solution of the Cauchy problem for the Benjamin-Ono equationu _t +uu _x −Hu _xx =0, whereH denotes the Hilbert transform. Our numerical method first approximates this Cauchy problem by an initial-value problem for a corresponding 2L-periodic problem in the spatial variable, withL large. This periodic problem is then solved using the Crank-Nicolson approximation in time and finite difference approximations in space, treating the nonlinear term in a standard conservative fashion, and the Hilbert transform by a quadrature formula which may be computed efficiently using the Fast Fourier Transform.     

Error estimates for a mixed finite element discretization of some degenerate parabolic equations

We consider a numerical scheme for a class of degenerate parabolic equations, including both slow and fast diffusion cases. A particular example in this sense is the Richards equation modeling the flow in porous media. The numerical scheme is based on the mixed finite element method (MFEM) in space, and is of one step implicit in time. The lowest order Raviart–Thomas elements are used. Here we extend the results in Radu et al. (SIAM J Numer Anal 42:1452–1478, 2004), Schneid et al. (Numer Math 98:353–370, 2004) to a more general framework, by allowing for both types of degeneracies. We derive error estimates in terms of the discretization parameters and show the convergence of the scheme. The features of the MFEM, especially of the lowest order Raviart–Thomas elements, are now fully exploited in the proof of the convergence. The paper is concluded by numerical examples.     

Numerical recovery of Robin boundary from boundary measurements for the Laplace equation

Based on an integral equation formulation, we present numerical methods for the inverse problem of recovering part of the domain boundary from boundary measurements of solutions to the Laplace equation on an accessible part of the boundary.     

The numerical analysis on a Volterra equation with asymptotically periodic solution

We consider order one operational quadrature methods on a certain integro-differential equation of Volterra type on (0,∞), with piecewise linear convolution kernels. The forms of discretization solution are patterned after a continuous one of Hannsgen (1979) [2]. An l¹ remainder stability and an error bound are derived.     

Lobatto IIIA-IIIB discretization of the strongly coupled nonlinear Schrödinger equation

In this paper, we construct a second order semi-explicit multi-symplectic integrator for the strongly coupled nonlinear Schrödinger equation based on the two-stage Lobatto IIIA-IIIB partitioned Runge-Kutta method. Numerical results for different solitary wave solutions including elastic and inelastic collisions, fusion of two solitons and with periodic solutions confirm the excellent long time behavior of the multi-symplectic integrator by preserving global energy, momentum and mass.     

A defect-correction method for unsteady conduction-convection problems II: Time discretization

In this paper, a fully discrete defect-correction mixed finite element method (MFEM) for solving the non-stationary conduction-convection problems in two dimension, which is leaded by combining the Back Euler time discretization with the two-step defect correction in space, is presented. In this method, we solve the nonlinear equations with an added artificial viscosity term on a finite element grid and correct these solutions on the same grid using a linearized defect-correction technique. The stability and the error analysis are derived. The theory analysis shows that our method is stable and has a good convergence property. Some numerical results are also given, which show that this method is highly efficient for the unsteady conduction-convection problems.     

On spectral methods for Volterra-type integro-differential equations

This paper considers the spectral methods for a Volterra-type integro-differential equation. Firstly, the Volterra-type integro-differential equation is equivalently restated as two integral equations of the second kind. Secondly, a Legendre-collocation method is used to solve them. Then the error analysis is conducted based on the _L∞$L_{\infty }$-norm. In addition, numerical results are presented to confirm our analysis.     

Numerical accuracy of real inversion formulas for the Laplace transform

In this article, we investigate and compare a number of real inversion formulas for the Laplace transform. The focus is on the accuracy and applicability of the formulas for numerical inversion. In this contribution, we study the performance of the formulas for measures concentrated on a positive half-line to continue with measures on an arbitrary half-line. As our trial measure concentrated on a positive half-line, we take the broad Gamma probability distribution family.     

Maximal regularity of type L p for abstract parabolic Volterra equations

We prove maximal regularity results of type L_p for abstract parabolic Volterra equations including problems with inhomogeneous boundary data. Our approach is purely operator theoretic. It uses the inversion of the convolution, the Dore-Venni theorem, the Mikhlin theorem in the operator-valued version, and real interpolation. Known results on L_p-regularity of abstract Cauchy problems and abstract parabolic pdes with inhomogeneous boundary conditions are recovered. As an application we consider the heat equation of memory type with inhomogeneous boundary condition.     

A combined finite volume–nonconforming/mixed-hybrid finite element scheme for degenerate parabolic problems

We propose and analyze a numerical scheme for nonlinear degenerate parabolic convection–diffusion–reaction equations in two or three space dimensions. We discretize the diffusion term, which generally involves an inhomogeneous and anisotropic diffusion tensor, over an unstructured simplicial mesh of the space domain by means of the piecewise linear nonconforming (Crouzeix–Raviart) finite element method, or using the stiffness matrix of the hybridization of the lowest-order Raviart–Thomas mixed finite element method. The other terms are discretized by means of a cell-centered finite volume scheme on a dual mesh, where the dual volumes are constructed around the sides of the original mesh. Checking the local Péclet number, we set up the exact necessary amount of upstream weighting to avoid spurious oscillations in the convection-dominated case. This technique also ensures the validity of the discrete maximum principle under some conditions on the mesh and the diffusion tensor. We prove the convergence of the scheme, only supposing the shape regularity condition for the original mesh. We use a priori estimates and the Kolmogorov relative compactness theorem for this purpose. The proposed scheme is robust, only 5-point (7-point in space dimension three), locally conservative, efficient, and stable, which is confirmed by numerical experiments.This work was supported by the GdR MoMaS, CNRS-2439, ANDRA, BRGM, CEA, EdF, France.     

Existence and uniqueness of viscosity solutions for a degenerate parabolic equation associated with the infinity-Laplacian

The existence, uniqueness and regularity of viscosity solutions to the Cauchy–Dirichlet problem are proved for a degenerate nonlinear parabolic equation of the form , where denotes the so-called infinity-Laplacian given by . To do so, a coercive regularization of the equation is introduced and barrier function arguments are also employed to verify the equi-continuity of approximate solutions. Furthermore, the Cauchy problem is also studied by using the preceding results on the Cauchy–Dirichlet problem. Dedicated to the memory of our friend Kyoji Takaichi. The research of the first author was partially supported by Waseda University Grant for Special Research Projects, #2004A-366.     

A second-order accurate numerical method for a fractional wave equation

We study a generalized Crank–Nicolson scheme for the time discretization of a fractional wave equation, in combination with a space discretization by linear finite elements. The scheme uses a non-uniform grid in time to compensate for the singular behaviour of the exact solution at t = 0. With appropriate assumptions on the data and assuming that the spatial domain is convex or smooth, we show that the error is of order k ² + h ², where k and h are the parameters for the time and space meshes, respectively.     

Relaxation of the curve shortening flow via the parabolic Ginzburg -Landau equation

In this paper we study how to find solutions to the parabolic Ginzburg–Landau equation that as have as interface a given curve that evolves under curve shortening flow. Moreover, for compact embedded curves we find a uniform profile for the solution up the extinction time of the curve. We show that after the extinction time the solution converges uniformly to a constant.     

Time discretization of an evolution equation via Laplace transforms

Following earlier work by Sheen, Sloan, and Thomée concerningparabolic equations we study the discretization in time of aVolterra type integro-differential equation in which the integraloperator is a convolution of a weakly singular function andan elliptic differential operator in space. The time discretizationis accomplished by using a modified Laplace transform in timeto represent the solution as an integral along a smooth curveextending into the left half of the complex plane, which isthen evaluated by quadrature. This reduces the problem to afinite set of elliptic equations with complex coefficients,which may be solved in parallel. Stability and error boundsof high order are derived for two different choices of the quadraturerule. The method is combined with finite-element discretizationin the spatial variables.     

On the Galerkin finite element approximations to multi-dimensional differential and integro-differential parabolic equations

The maximum norm error estimates of the Galerkin finite element approximations to the solutions of differential and integro-differential multi-dimensional parabolic problems are considered. Our method is based on the use of the discrete version of the elliptic-Sobolev inequality and some operator representations of the finite element solutions. The results of the present paper lead to the error estimates of optimal or almost optimal order for the case of simplicial Lagrangian piecewise polynomial elements.     

Toeplitz matrix method and nonlinear integral equation of Hammerstein type

The existence and uniqueness solution of the nonlinear integral equation of Hammerstein type with discontinuous kernel are discussed. The normality and continuity of the integral operator are proved. Toeplitz matrix method is used, as a numerical method, to obtain a nonlinear system of algebraic equations. Also, many important theorems related to the existence and uniqueness of the produced algebraic system are derived. Finally, numerical examples, when the kernel takes a logarithmic and Carleman forms, are discussed and the estimate error, in each case, is calculated.