Local Boundary Value Problem Research Articles

Analysis and applications of a generalized finite element method with global–local enrichment functions

This paper presents a procedure to build enrichment functions for partition of unity methods like the generalized finite element method and the hp cloud method. The procedure combines classical global–local finite element method concepts with the partition of unity approach. It involves the solution of local boundary value problems using boundary conditions from a global problem defined on a coarse discretization. The local solutions are in turn used to enrich the global space using the partition of unity framework. The computations at local problems can be parallelized without difficulty allowing the solution of large problems very efficiently.The effectiveness of the approach in terms of convergence rates and computational cost is investigated in this paper. We also analyze the effect of inexact boundary conditions applied to local problems and the size of the local domains on the accuracy of the enriched global solution.Key aspects of the computational implementation, in particular, the numerical integration of generalized FEM approximations built with global–local enrichment functions, are presented.The method is applied to fracture mechanics problems with multiple cracks in the domain. Our numerical experiments show that even on a serial computer the method is very effective and allows the solution of complex problems. Our analysis also demonstrates that the accuracy of a global problem defined on a coarse mesh can be controlled using a fixed number of global degrees of freedom and the proposed global–local enrichment functions.

Local boundary value problems for the error in FE approximation of non‐linear diffusion systems

AbstractIn this work we investigate the a posteriori error estimation for a class of non‐linear, multicomponent diffusion operators, which includes the Stefan–Maxwell equations. The local error indicators for the global error are based on local boundary value problems, which are chosen to approximate either the global residual of the finite element approximation or the global linearized error equation. Using representative numerical examples, it is shown that the error indicators based on the latter approach are more accurate for estimating the global error for this problem class as the problem becomes more non‐linear, and can even produce better adaptive mesh refinement (AMR). In addition, we propose a new local error indicator for the error in output functionals that is accurate, inexpensive to compute, and is suitable for AMR, as demonstrated by numerical examples. Copyright © 2007 John Wiley & Sons, Ltd.

A kernel-independent adaptive fast multipole algorithm in two and three dimensions

We present a new fast multipole method for particle simulations. The main feature of our algorithm is that it does not require the implementation of multipole expansions of the underlying kernel, and it is based only on kernel evaluations. Instead of using analytic expansions to represent the potential generated by sources inside a box of the hierarchical FMM tree, we use a continuous distribution of an equivalent density on a surface enclosing the box. To find this equivalent density, we match its potential to the potential of the original sources at a surface, in the far field, by solving local Dirichlet-type boundary value problems. The far-field evaluations are sparsified with singular value decomposition in 2D or fast Fourier transforms in 3D. We have tested the new method on the single and double layer operators for the Laplacian, the modified Laplacian, the Stokes, the modified Stokes, the Navier, and the modified Navier operators in two and three dimensions. Our numerical results indicate that our method compares very well with the best known implementations of the analytic FMM method for both the Laplacian and modified Laplacian kernels. Its advantage is the (relative) simplicity of the implementation and its immediate extension to more general kernels.

An investigation of overlapping domain decomposition methods for one-dimensional dispersive long wave equations

This paper contains a thorough investigation of the numerical accuracy and efficiency of an overlapping domain decomposition (Schwarz iteration) method for weakly dispersive and non-linear water waves. The investigation is restricted to one-dimensional wave propagation. In our tests, a global domain is divided into two overlapping subdomains. A local boundary value problem (at a time level) is solved in each subdomain, with boundary values extracted from the neighboring domains. The size of the overlap and number of iterations over the domains are the principal parameters that influence the convergence speed and numerical accuracy. We also investigate different finite difference and finite element formulations. The finite signal speed in wave problems makes domain decomposition methods particularly efficient, and the proposed method converges satisfactorily already for an overlap of some typical water depths and a few (2–3) iterations. Numerical artifacts and instabilities may develop at the subdomain boundaries for certain choices of overlap, wave lengths, grid sizes, and (discrete) wave velocities. A special filtering technique is designed to make the method more robust with respect to such instabilities.

A posteriori error estimates for a mixed-FEM formulation of a non-linear elliptic problem

We consider the numerical solution, via the mixed finite element method, of a non-linear elliptic partial differential equation in divergence form with Dirichlet boundary conditions. Besides the temperature u and the flux σ , we introduce ∇ u as a further unknown, which yields a variational formulation with a twofold saddle point structure. We derive a reliable a posteriori error estimate that depends on the solution of a local linear boundary value problem, which does not need any equilibrium property for its solvability. In addition, for specific finite element subspaces of Raviart–Thomas type we are able to provide a fully explicit a posteriori error estimate that does not require the solution of the local problems. Our approach does not need the exact finite element solution, but any reasonable approximation of it, such as, for instance, the one obtained with a fully discrete Galerkin scheme. In particular, we suggest a scheme that uses quadrature formulas to evaluate all the linear and semi-linear forms involved. Finally, several numerical results illustrate the suitability of the explicit error estimator for the adaptive computation of the corresponding discrete solutions.

A High-Order Difference Scheme for a Nonlocal Boundary-Value Problem for the Heat Equation

Abstract This paper is concerned with a high order difference scheme for a non- local boundary-value problem of parabolic equation. The integrals in the boundary equations are approximated by the composite Simpson rule. The unconditional solv- ability and L_∞ convergence of the difference scheme is proved by the energy method. The convergence rate of the difference scheme is second order in time and fourth order in space. Some numerical examples are provided to illustrate the convergence.

Humidity transfer in unsaturated heterogeneous porous media by homogenization

In the paper the one-equation model of humidity transfer in unsaturated macroscopically heterogeneous porous media is presented. The homogenization method by two-scale asymptotic expansions is used to derive the upscaled form of the Richard equation, which is commonly used when the medium is considered as macroscopically homogeneous. This equation is highly non-linear due to the pressure-dependence of the hydrodynamic characteristics of the porous medium. The domain of validity of the model is explicitly given, namely: the length-scales separation, the characteristic time scale condition and the ratio of the hydrodynamic characteristics being of the same orders of magnitude. The effective capillary capacity and the effective hydraulic conductivity for an equivalent continuum are defined in terms of geometry and local hydrodynamic characteristics of the porous medium. A procedure of determination of the effective suction curve and the effective hydraulic conductivity curve as functions of the average water content for any type of the macroscopic heteregeneity for which the method can be applied, is provided. Since the problem is non-linear this procedure involves the resolution of a local boundary value problem formulated over a period for each value of suction. In two or three-dimensional cases, this problem can be solved using the numerical methods for any geometry of the medium. In a one-dimensional case it was shown that the analytical solution gives the well-known results of harmonic and arithmetic mean.

Field stitching algorithm for the analysis of electrically large diffractive optical elements

A rigorous electromagnetic algorithm is presented for the analysis of electrically large diffractive optical elements (DOE's), i.e., those that contain small features and have large apertures compared with the wavelength of illumination. The technique uses a finite-sized analysis window within which a rigorous electromagnetic technique is used to solve the local boundary-value problem. To this end the boundary-element and finite-difference time-domain methods are used. The analysis window is translated over the entire surface of the DOE and stitches together the complete solution. We validate the techniques by comparing the stitched boundary fields with those of a complete analysis, in both magnitude and phase, for a binary lens. To illustrate the utility of our method we analyzed an eight-level diffractive lens with a 10, 000-wavelength diameter sampled at 0.05 wavelength that required 8 Mbytes of memory on a desktop personal computer.

Higher-Codimensional Boundary Value Problems and $F$-Mild Microfunctions —Local and Microlocal Uniqueness—

For the study of local and microlocal boundary value problems with a boundary of codimension greater than one, sheaves of F -mild hyperfunctions and F -mild microfunctions are introduced. They are refinements of the notions of hyperfunctions and microfunctions with real analytic parameters and have natural boundary values. F -mild solutions of a general \mathcal D -Module \mathcal M (that is, a system of linear partial differential equations with analytic coefficients) are considered. In particular, local and microlocal uniqueness in the boundary value problem (the Holmgren type theorem) is proved if the boundary is non-characteristic for \mathcal M . or else if \mathcal M is Fuchsian along the boundary.

Analysis and adaptive modeling of highly heterogeneous elastic structures

In this paper, we develop a theory and methodology for obtaining approximate solutions to boundary value problems describing the deformation of highly heterogeneous linearly-elastic structures. The method, which represents a significant departure from traditional homogenization methods, provides a systematic and rigorous approach towards resolving the effects of microstructure of different scales on the macroscopic response of complex heterogeneous structures. An early variant of this method was first introduced in [10]. The method, referred to as HDPM (Homogenized Dirichlet Projection Method), proceeds by first solving an auxiliary homogenized boundary value problem, describing the deformation of a structure with the same exterior geometry, but with a selected set of uniform material properties. The adequacy of this homogenized solution is then determined using an a posteriori error estimate that provides a measure of the error of the homogenized solution in subdomains of the structure compared to the solution of the fine-scale heterogeneous problem, without specific knowledge of this fine-scale solution. In those subdomains where the homogenized solution is deemed acceptable, it is retained. However, in subdomains where the homogenized solution is inadequate, as is determined when the estimated error exceeds a preset tolerance, a local boundary value problem is constructed by projecting the homogenized displacements onto a partition designed to isolate these subdomains. These local boundary value problems are then solved in those subdomains where the homogenized solution is inaccurate using the exact microstructure with the approximate local boundary conditions. A posteriori error estimates are made to ascertain the quality of the resulting solution. If the quality of the solution remains inadequate, above a preset error tolerance, a two stage adaptive procedure is implemented. Stage-I (‘material adaptivity’) corresponds to modifying the homogenized structure's material properties. If, after Stage I, the solution quality is still inadequate, the subdomains of local solution are enlarged, thereby modeling in greater detail the actual microstructure, and the local solution process is repeated (Stage-II, ‘subdomain unrefinement’). The main feature of this method is that only in subdomains where the error in the usual homogenized solution is above a preset tolerance is the microstructure taken into account. The cost of this method is shown to be orders of magnitude cheaper than direct large-scale computational simulations of micromechanical events. The results of several numerical experiments are provided to demonstrate the method and verify theoretical estimates.

Effective diffusion coefficient : From homogenization to experiment

In this paper, an example of the application of the homogenization approach (asymptotic expansion technique) to predict the effective diffusion coefficient for an equivalent continuum, together with the experimental verification of the theoretical results is presented. The experimental setup was constructed for the measurements of diffusion in a model periodic porous medium made of Plexiglas. The computer program using the FEM was elaborated to solve the local boundary value problem for a period and to calculate the effective diffusion coefficient. The comparison between the theory and the experiment indicates good agreement between the numerical and experimental values of the effective diffusion coefficient. Interpretation of the test data from the point of view of the homogenization theory is also incorporated.

A FUNCTION THEORY METHOD IN ELLIPTIC PROBLEMS IN THE PLANE. II. THE PIECEWISE SMOOTH CASE

A general boundary value problem, encompassing from a unified viewpoint a broad circle of local and nonlocal boundary value problems, is studied for elliptic systems with real, constant (and only leading) matrix coefficients. A method is given for the equivalent reduction of this problem to a system of boundary equations. The considerations are carried out in domains with piecewise smooth boundaries and in weighted spaces. A Noetherian criterion and an index formula for this problem are established, and the asymptotics of its solution in a neighborhood of corner points is described.

A FUNCTION THEORY METHOD IN BOUNDARY VALUE PROBLEMS IN THE PLANE. I: THE SMOOTH CASE

A general (not necessarily local) boundary value problem is considered for an elliptic system on the plane of th order containing only leading terms with constant coefficients. By a method of function theory developed for elliptic systems of first order with a constant triangular matrix , ; this problem is reduced to an equivalent system of integrofunctional equations on the boundary. In particular, a criterion that the problem be Noetherian and a formula for its index are obtained in this way. All considerations are carried out in the smooth case when the boundary of the domain has no corner points, while the boundary operators act in spaces of continuous functions.

A numerical method for solving interface problems arising in two-point boundary value problems

A numerical approach iterating on the position of interface points is suggested for solving interface problems arising in two-point boundary value problems. Given an interface problem, it is decomposed into several standard local boundary value problems which are coupled at the interface points. Meanwhile, a nonlinear equation involving the interface points are formulated from the interface conditions. The advantages of this approach are that the boundary conditions for each local problem can easily be selected by considering the natural physical requirements, and that each of the local problems can be solved independently by standard numerical BVP techniques.

Lifting of local subdifferentiations and elliptic boundary value problems on symmetric domains, I

Lifting of local subdifferentiations and elliptic boundary value problems on symmetric domains, I

Lifting of local subdifferentiations and elliptic boundary value problems on symmetric domains, II

On etudie le probleme aux valeurs propres non lineaire -Δu=λ exp(u) (dans Ω), u=0 (sur ∂Ω) pour λ sur Ω={x|a<|x|<1}⊂R 2 , ou 0<a<1. A partir d'une methode variationnelle on montre l'existence de solutions non radiales multiples

Local error estimates for adaptive mesh refinement

Five methods for local error estimation in a finite-element solution for adaptive meshing are analyzed. All these methods are a-posteriori, single-solution, element-by-element error estimators. Two are based on the estimation of the first neglected term in the series expansion of the solution, a third is based on the approximate solution of a local boundary-value problem with hierarchical shape functions, and the other two utilize a residual evaluation. All the methods are described and implemented in a common environment, and the results obtained in a simple test case are presented and discussed. >

On a free boundary value problem of physical geodesy, II (existence)

AbstractThe determination of the figure of the earth is considered as a local free boundary value problem of potential theory. In the C1,ε‐topology local existence is proved by means of Legendre transforms. This method also provides an elementary argument for the regularity of the solutions.

Some local and nonlocal boundary value problems for hyperbolic equations (in Russian)

The paper deals with local and nonlocal boundary value problems for hyperbolic equations, which generalize the Darboux, Goursat and Cauchy problems. Under certain conditions on the given functions the correctness of these problems is proved, in other cases the uniqueness of the solution is proved and theorems of uniqueness of the solution in a more wide class of functions are stated. In the paper examples are given, showing the non-stability of the solution if some of the conditions on the boundary data are violated.

On a free boundary value problem of physical geodesy, I (uniqueness)

AbstractThe determination of the figure of the earth is considered as a local free boundary value problem of potential theory: the shape of a slowly rotating heavy body has to be found from the boundary data of the attracting force and its potential outside of the body, provided that an approximation to the solution is already known. In this paper a uniqueness result is proved which is local with respect to the C1‐topology.