Showing papers in "Numerical Methods for Partial Differential Equations in 2007"
TL;DR: In this paper, the steady state fractional advection dispersion equation (FADE) on bounded domains in ℝd is discussed and a theoretical framework for the variational solution of FADE is presented.
Abstract: In this article, we discuss the steady state fractional advection dispersion equation (FADE) on bounded domains in ℝd. Fractional differential and integral operators are defined and analyzed. Appropriate fractional derivative spaces are defined and shown to be equivalent to the fractional dimensional Sobolev spaces. A theoretical framework for the variational solution of the steady state FADE is presented. Existence and uniqueness results are proven, and error estimates obtained for the finite element approximation. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 256–281, 2007
239 citations
TL;DR: It is shown that the approximate solution exists and is unique, which is not obvious since the scheme is nonlinear, and it is proved that, for general W−1,p′(Ω) source term and W1‐(1/p),p(∂ Ω) boundary data, the approximate Solution and its discrete gradient converge strongly towards the exact solution and its gradient, respectively, in appropriate Lebesgue spaces.
Abstract: Discrete duality finite volume schemes on general meshes, introduced by Hermeline and Domelevo and Omnes for the Laplace equation, are proposed for nonlinear diffusion problems in 2D with nonhomogeneous Dirichlet boundary condition. This approach allows the discretization of non linear fluxes in such a way that the discrete operator inherits the key properties of the continuous one. Furthermore, it is well adapted to very general meshes including the case of nonconformal locally refined meshes. We show that the approximate solution exists and is unique, which is not obvious since the scheme is nonlinear. We prove that, for general W−1,p′(Ω) source term and W1-(1/p),p(∂Ω) boundary data, the approximate solution and its discrete gradient converge strongly towards the exact solution and its gradient, respectively, in appropriate Lebesgue spaces. Finally, error estimates are given in the case where the solution is assumed to be in W2,p(Ω). Numerical examples are given, including those on locally refined meshes. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007
164 citations
TL;DR: In this article, the authors apply the reduced basis method to solve Navier-Stokes equations in parametrized domains, including the case of nonaffine parametric dependence that is treated by an empirical interpolation method.
Abstract: We apply the reduced basis method to solve Navier-Stokes equations in parametrized domains. Special attention is devoted to the treatment of the parametrized nonlinear transport term in the reduced basis framework, including the case of nonaffine parametric dependence that is treated by an empirical interpolation method. This method features (i) a rapid global convergence owing to the property of the Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N (optimally) selected points in the parameter space, and (ii) the offline/online computational procedures that decouple the generation and projection stages of the approximation process. This method is well suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control. Our analysis focuses on: (i) the pressure treatment of incompressible Navier-Stokes problem; (ii) the fulfillment of an equivalent inf-sup condition to guarantee the stability of the reduced basis solutions. The applications that we consider involve parametrized geometries, like e.g. a channel with curved upper wall or an arterial bypass configuration.
155 citations
TL;DR: In this article, for differential equations modeling systems where the solutions satisfy a positivity condition, procedures can be formulated to calculate the so-called denominator functions that appear in the discrete derivatives.
Abstract: An essential feature of nonstandard finite difference schemes for differential equations is the precise manner in which the discretization of derivatives is made. We demonstrate, for differential equations modeling systems where the solutions satisfy a positivity condition, that procedures can be formulated to calculate the so-called denominator functions that appear in the discrete derivatives. These procedures are applied to a number of both ordinary and partial model differential equations to illustrate their use. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007
155 citations
TL;DR: In this paper, the authors survey the Trefftz method (TM), the collocation method (CM), and the collocated Treffitz method (CTM) and conclude that the CTM is the simplest algorithm and provides the most accurate solution with the best numerical stability.
Abstract: In this article we survey the Trefftz method (TM), the collocation method (CM), and the collocation Trefftz method (CTM). We also review the coupling techniques for the interzonal conditions, which include the indirect Trefftz method, the original Trefftz method, the penalty plus hybrid Trefftz method, and the direct Trefftz method. Other boundary methods are also briefly described. Key issues in these algorithms, including the error analysis, are addressed. New numerical results are reported. Comparisons among TMs and other numerical methods are made. It is concluded that the CTM is the simplest algorithm and provides the most accurate solution with the best numerical stability. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007
105 citations
TL;DR: In this paper, a numerical technique is developed for the one-dimensional hyperbolic equation that combine classical and integral boundary conditions, which is based on shifted Legendre tau technique, and illustrative examples are included to demonstrate the validity and applicability of the presented technique.
Abstract: The hyperbolic partial differential equation with an integral condition arises in many physical phenomena. In this research a numerical technique is developed for the one-dimensional hyperbolic equation that combine classical and integral boundary conditions. The proposed method is based on shifted Legendre tau technique. Illustrative examples are included to demonstrate the validity and applicability of the presented technique. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 282–292, 2007
76 citations
TL;DR: A modified and more accurate discretization is derived that gives second‐order convergence of the fluid velocity and the stress of the solid and is proven for the case when the interface coincides with a grid node.
Abstract: Finite difference discretizations of 1D poroelasticity equations with discontinuous coefficients are analyzed. A recently suggested FD discretization of poroelasticity equations with constant coefficients on staggered grid, [5], is used as a basis. A careful treatment of the interfaces leads to harmonic averaging of the discontinuous coefficients. Here, convergence for the pressure and for the displacement is proven in certain norms for the scheme with harmonic averaging (HA). Order of convergence 1.5 is proven for arbitrary located interface, and second order convergence is proven for the case when the interface coincides with a grid node. Furthermore, following the ideas from [3], modified HA discretization are suggested for particular cases. The velocity and the stress are approximated with second order on the interface in this case. It is shown that for wide class of problems, the modified discretization provides better accuracy. Second order convergence for modified scheme is proven for the case when the interface coincides with a displacement grid node. Numerical experiments are presented in order to illustrate our considerations.
65 citations
TL;DR: In this article, a finite element formulation for coupled flow and geomechanics is presented, where pressure and displacement solutions are staggered during a time step until a convergence tolerance is satisfied.
Abstract: We present a finite element formulation for coupled flow and geomechanics. We use mixed finite element spaces to approximate pressure and continuous Galerkin methods for displacements. In solving the coupled system, pressure and displacements can be solved either simultaneously in a fully coupled scheme or sequentially in a loosely coupled scheme. In this paper we formulate an iterative method where pressure and displacement solutions are staggered during a time step until a convergence tolerance is satisfied. A priori convergence results for the iterative coupling are also presented, along with a summary of the convergence results for the fully coupled scheme. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 785-797, 2007
57 citations
TL;DR: In this paper, the usual linear elastodynamics equations augmented with evolution equations for viscoelastic internal stresses are considered and a fully discrete approximation is defined, based on a spatially symmetric or non-symmetric interior penalty discontinuous Galerkin finite element method, and a displacement-velocity centred difference time discretisation.
Abstract: We consider the usual linear elastodynamics equations augmented with evolution equations for viscoelastic internal stresses. A fully discrete approximation is defined, based on a spatially symmetric or non-symmetric interior penalty discontinuous Galerkin finite element method, and a displacement-velocity centred difference time discretisation. An a priori error estimate is given but only the main ideas in the proof of the error estimate are reported here due to the large number of (mostly technical) estimates that are required. The full details are referenced to a technical report. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007
54 citations
TL;DR: The mathematical models describing such problems are often based on a relatively small amount of available information as mentioned in this paper, which makes it difficult to simulate large, complex physical problems on a large number of computers.
Abstract: Today's computers allow us to simulate large, complex physical problems. Many times the mathematical models describing such problems are based on a relatively small amount of available information ...
52 citations
TL;DR: This article explores the linear algebraic aspects of the multiscale algorithm, showing that it involves a blend of both classical overlapping Schwarz methods and nonoverlapping Schur methods, and extends the algorithm and the theory from its additive variant to obtain new hybrid and deflation variants.
Abstract: In this article, we describe a new class of domain decomposition preconditioners suitable for solving elliptic PDEs in highly fractured or heterogeneous media, such as arise in groundwater flow or oil recovery applications. Our methods employ novel coarsening operators, which are adapted to the heterogeneity of the media. In contrast to standard methods (based on piecewise polynomial coarsening), the new methods can achieve robustness with respect to coefficient discontinuities even when these are not resolved by a coarse mesh. This situation arises often in practical flow computation, in both the deterministic and (Monte-Carlo simulated) stochastic cases. An example of a suitable coarsener is provided by multiscale finite elements. In this article, we explore the linear algebraic aspects of the multiscale algorithm, showing that it involves a blend of both classical overlapping Schwarz methods and nonoverlapping Schur methods. We also extend the algorithm and the theory from its additive variant to obtain new hybrid and deflation variants. Finally, we give extensive numerical experiments on a range of heterogeneous media problems illustrating the properties of the methods. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 859-878, 2007
TL;DR: In this paper, a numerical technique is presented for the solution of a nonclassical problem for the one-dimensional wave equation, which uses the cubic B-spline scaling functions.
Abstract: Problems for parabolic partial differential equations with nonlocal boundary conditions have been studied in many articles, but boundary value problems for hyperbolic partial differential equations have so far remained nearly uninvestigated. In this article a numerical technique is presented for the solution of a nonclassical problem for the one-dimensional wave equation. This method uses the cubic B-spline scaling functions. Some numerical results are reported to support our study. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007
TL;DR: In this article, a Crank-Nicolson-type finite difference scheme for the nonlinear evolutionary Cahn-Hilliard equation is presented and its existence, uniqueness and convergence is proved.
Abstract: In this article, we analyze a Crank-Nicolson-type finite difference scheme for the nonlinear evolutionary Cahn-Hilliard equation. We prove existence, uniqueness and convergence of the difference solution. An iterative algorithm for the difference scheme is given and its convergence is proved. A linearized difference scheme is presented, which is also second-order convergent. Finally a new difference method possess a Lyapunov function is presented. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 437–455, 2007
TL;DR: In this article, the authors developed a set of compact difference schemes to solve the heat-conduction problem with Neumann boundary conditions, which is proved to be globally solvable and unconditionally stable.
Abstract: Inthisarticle,asetoffourth-ordercompactfinitedifferenceschemesisdevelopedtosolveaheatconductionproblem with Neumann boundary conditions. It is derived through the compact difference schemes at allinterior points, and the combined compact difference schemes at the boundary points. This set of schemesis proved to be globally solvable and unconditionally stable. Numerical examples are provided to verify theaccuracy.
TL;DR: One‐dimensional integrated RBFNs are employed to represent the variable along each line of the grid, resulting in a significant improvement of computational efficiency, and normal derivative boundary conditions are imposed by means of integration constants.
Abstract: []: This paper reports a new Cartesian-grid collocation method based on radial-basis-function
networks (RBFNs) for numerically solving elliptic partial
differential equations (PDEs) in irregular domains. The domain of interest is embedded in a Cartesian grid, and the governing equation is discretized by using a collocation approach. The new features here are (a) One-dimensional integrated RBFNs are employed to represent the variable along each line of the grid,
resulting in a significant improvement of computational
efficiency, (b) The present method does not require complicated interpolation techniques for the treatment of Dirichlet boundary conditions in order to achieve a high level of accuracy, and (c) Normal derivative boundary conditions are imposed by means of integration constants. The method is verified through the solution
of second- and fourth-order PDEs; accurate results and fast
convergence rates are obtained.
TL;DR: In this article, alternating-direction explicit (A.D.E) finite-difference methods are used for solving time-dependent advection-diffusion equations, and their stability characteristics are discussed.
Abstract: Alternating-Direction Explicit (A.D.E.) finite-difference methods make use of two approximations that are implemented for computations proceeding in alternating directions, e.g., from left to right and from right to left, with each approximation being explicit in its respective direction of computation. Stable A.D.E. schemes for solving the linear parabolic partial differential equations that model heat diffusion are wellknown, as are stable A.D.E. schemes for solving the first-order equations of fluid advection. Several of these are combined here to derive A.D.E. schemes for solving time-dependent advection-diffusion equations, and their stability characteristics are discussed. In each case, it is found that it is the advection term that limits the stability of the scheme. The most stable of the combinations presented comprises an unconditionally stable approximation for computations carried out in the direction of advection of the system, from left to right in this case, and a conditionally stable approximation for computations proceeding in the opposite direction. To illustrate the application of the methods and verify the stability conditions, they are applied to some quasi-linear one-dimensional advection-diffusion problems. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 00: 000–000, 2007
TL;DR: The goals of this work are to provide a precise mathematical framework for the real-time finite element solution of the problems of calibration, optimal heat source control, and goal-oriented error estimation applied to the equations of bioheat transfer and demonstrate that current finite element technology, parallel computer architecture, data transfer infrastructure, and thermal imaging modalities are capable of inducing a precise computer controlled temperature field within the biological domain.
Abstract: Elevating the temperature of cancerous cells is known to increase their susceptibility to subsequent radiation or chemotherapy treatments, and in the case in which a tumor exists as a well-defined region, higher intensity heat sources may be used to ablate the tissue. These facts are the basis for hyperthermia based cancer treatments. Of the many available modalities for delivering the heat source, the application of a laser heat source under the guidance of real-time treatment data has the potential to provide unprecedented control over the outcome of the treatment process [7, 18]. The goals of this work are to provide a precise mathematical framework for the real-time finite element solution of the problems of calibration, optimal heat source control, and goal-oriented error estimation applied to the equations of bioheat transfer and demonstrate that current finite element technology, parallel computer architecture, data transfer infrastructure, and thermal imaging modalities are capable of inducing a precise computer controlled temperature field within the biological domain.
TL;DR: An a posteriori error estimator for expanded mixed hybrid finite‐element methods for second‐order elliptic problems yields reliable and efficient estimate based on residuals.
TL;DR: A 9‐point fourth‐order finite difference scheme for 2D elliptic problems with a mixed derivative and variable coefficients is proposed and a class of two‐level high‐order compact schemes with weighted time discretization is derived.
Abstract: We propose a 9-point fourth-order finite difference scheme for 2D elliptic problems with a mixed derivative and variable coefficients. The same approach is extended to derive a class of two-level high-order compact schemes with weighted time discretization for solving 2D parabolic problems with a mixed derivative. The schemes are fourth-order accurate in space and second- or lower-order accurate in time depending on the choice of a weighted average parameter μ. Unconditional stability is proved for 0.5 ≤ μ ≤ 1, and numerical experiments supporting our theoretical analysis and confirming the high-order accuracy of the schemes are presented. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 366–378, 2007
TL;DR: In this paper, a classification of fitted mesh methods for time-dependent reaction diusion problems is presented, based on the analyses of Lins [4] for the analogous steady-state problem and of Kopteva [3] of timedependent convection-diusi on problems.
Abstract: Using discrete Green’s functions techniques, we present a classification of fitted mesh methods for time-dependent reaction diusion problems, based on the analyses of Lins [4] for the analogous steady-state problem and of Kopteva [3] of time-dependent convection-diusi on problems. As examples of how to apply the analysis, we derive error estimates for the fitted meshes of Shishkin and Bakhvalov, and provide supporting numerical results.
TL;DR: In this article, a new family of numerical schemes for inhomogeneous parabolic partial differential equations was developed utilizing diagonal Pade schemes combined with positivity-preserving Pade scheme as damping devices.
Abstract: A new family of numerical schemes for inhomogeneous parabolic partial differential equations is developed utilizing diagonal Pade schemes combined with positivity–preserving Pade schemes as damping devices. We also develop a split version of the algorithm using partial fraction decomposition to address difficulties with accuracy and computational efficiency in solving and to implement the algorithms in parallel. Numerical experiments are presented for several inhomogeneous parabolic problems, including pricing of financial options with nonsmooth payoffs.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007
TL;DR: In this article, the authors considered an inverse problem for estimating the two coefficient functions c and k in a parabolic type partial differential equation with the aid of the measurements of u at two different times.
Abstract: We consider an inverse problem for estimating the two coefficient functions c and k in a parabolic type partial differential equation c(u)ut = ∂[k(u)ux]/∂x with the aid of the measurements of u at two different times. The first- and second-order one-step group preserving schemes are developed. Solving the resultant algebraic equations with a closed-form, we can estimate the unknown temperature-dependent thermal conductivity and heat capacity. The new methods possess threefold advantages: they do not require any a priori information on the functional forms of thermal conductivity and heat capacity; no initial guesses are required; and no iterations are required. Numerical examples are examined to show that the new approaches have high accuracy and efficiency, even there are rare measured data. When the measured temperatures are polluted by uniform or normal random noise, the estimated results are also good. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007
TL;DR: This work considers a convection–diffusion problem with Dirichlet boundary conditions posed on a unit square and proves uniform convergence (in the perturbation parameter) in an associated norm and establishes a supercloseness result.
Abstract: We consider a convection–diffusion problem with Dirichlet boundary conditions posed on a unit square. The problem is discretized using a combination of the standard Galerkin FEM and an h–version of the nonsymmetric discontinuous Galerkin FEM with interior penalties on a layer–adapted mesh with linear/bilinear elements. With specially chosen penalty parameters for edges from the coarse part of the mesh, we prove uniform convergence (in the perturbation parameter) in an associated norm. In the same norm we also establish a supercloseness result. Numerical tests support our theoretical estimates.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007
TL;DR: Using nonpolynomial cubic spline in space and finite difference in time directions, this paper obtained the implicit three level methods of O(k2 + h2) and O (k 2 + h4).
Abstract: In this study, we developed the methods based on nonpolynomial cubic spline for numerical solution of second-order nonhomogeneous hyperbolic partial differential equation. Using nonpolynomial cubic spline in space and finite difference in time directions, we obtained the implicit three level methods of O(k2 + h2) and O(k2 + h4). The proposed methods are applicable to the problems having singularity at x = 0, too. Stability analysis of the presented methods have been carried out. The presented methods are applied to the nonhomogeneous examples of different types. Numerical comparison with Mohanty's method (Mohanty, Appl Math Comput, 165 (2005), 229–236) shows the superiority of our presented schemes. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007
TL;DR: This article considers the numerical solution of the time‐dependent Schrödinger equation in ℝ3 and demonstrates that the method given is effective and feasible.
Abstract: We consider the numerical solution of the time-dependent Schrodinger equation in ℝ3. An artificial boundary is introduced to obtain a bounded computational domain. On the given artificial boundary the exact boundary condition and a series of approximating boundary conditions are constructed, which are called artificial boundary conditions. By using the exact or approximating boundary conditions on the artificial boundary, the original problem is reduced to an equivalent or an approximate initial-boundary value problem on the bounded computational domain. The uniqueness of the approximate problem is proved. The numerical results demonstrate that the method given in this article is effective and feasible. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007
TL;DR: This article considers the extension of well‐known discontinuous Galerkin (DG) finite element formulations to elliptic problems with periodic boundary conditions and proposes an approach in which the periodicity constraint is incorporated weakly in the variational formulation of the problem.
Abstract: This article considers the extension of well-known discontinuous Galerkin (DG) finite element formulations to elliptic problems with periodic boundary conditions. Such problems routinely appear in a number of applications, particularly in homogenization of composite materials. We propose an approach in which the periodicity constraint is incorporated weakly in the variational formulation of the problem. Both H 1 and L2 error estimates are presented. A numerical example confirming theoretical estimates is shown. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 000–000, 2007