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The authors' objective in this paper is to find the numerical solutions of obstacle, unilateral and contact second‐order boundary‐value problems.

To achieve this, the authors formulate a spatially adaptive grid refinement scheme following Galerkin's finite element method based on a weighted‐residual. A residual based a‐posteriori error estimation scheme has been utilized for checking the approximate solutions for various finite element grids. The local element balance has been considered as an error assessment criterion. The approach utilizes piece‐wise linear approximations utilizing linear Langrange polynomials. Numerical experiments indicate that local errors are large in regions where the gradients are large.

A comparison of the spatially adaptive grid refinement with that of uniform meshing for second order obstacle boundary value problems confirms the superiority of the scheme without increasing the number of unknown coefficients.

The authors believe the work has merit not only in terms of the approach but also of the problem solved in the paper.

A flux‐upwind finite element method is developed for the carrier transport equations in semiconductor device modeling. Our approach is motivated by the streamline upwind methods that have proven effective in fluid mechanics. The procedure reduces precisely to the Scharfetter‐Gummel approach in one dimension. In higher‐dimensions, however, it differs from this classical technique and is shown here to generate more accurate solutions with less numerical dissipation. Numerical results are presented for representative MOSFET and p‐n junction devices to illustrate this point. Both upwind techniques have been implemented in conjunction with an adaptive finite element refinement procedure for better layer resolution and yield a more stable algorithm.

The Motz problem can be considered as a benchmark problem for testing the performance of numerical methods in the solution of elliptic problems with boundary singularities. The purpose of this paper is to address the solution of the Motz problem using the radial basis function (RBF) method, which is a truly meshfree scheme.

Both the global RBF collocation method (also known as Kansa's method) and the recently proposed local RBF‐based differential quadrature (LRBFDQ) method are considered. In both cases, it is shown that the accuracy of the solution can be significantly increased by using special functions which capture the behavior of the singularity. In the case of global collocation, the functional space spanned by the RBF is enlarged by adding singular functions which capture the behavior of the local singular solution. In the case of local collocation, the problem is modified appropriately in order to eliminate the singularities from the formulation.

The paper shows that the exponential convergence both with increasing resolution and increasing shape parameter, which is typical of the RBF method, is lost in problems containing singularities. The accuracy of the solution can be increased by collocation of the partial differential equation (PDE) at boundary nodes. However, in order to restore the exponential convergence of the RBF method, it is necessary to use special functions which capture the behavior of the solution near the discontinuity.

The paper uses Motz's problem as a prototype for problems described by elliptic partial differential equations with boundary singularities. However, the results obtained in the paper are applicable to a wide range of problems containing boundaries with conditions which change from Dirichlet to Neumann, thus leading to singularities in the first derivatives.

The paper shows that both the global RBF collocation method and the LRBFDQ method, are truly meshless methods which can be very useful for the solution of elliptic problems with boundary singularities. In particular, when complemented with special functions that capture the behavior of the solution near the discontinuity, the method exhibits exponential convergence both with resolution and with shape parameter.

A new point creation scheme is presented for generating unstructured uniform size two‐dimensional triangular meshes using the Delaunay triangulation method. The scheme is shown to be suitable for generating meshes with strict limits on element size, such as that used in acoustic and electromagnetic simulations. In this approach for generating meshes of an exact element size a grid made of regular triangles is used to cover most of the mesh domain. An advanced boundary refinement and alignment procedure is presented for optimal connection of the regular grid with the boundary. A new refinement scheme treats the reconnection triangles, between the regular grid and boundary elements, until all are judged satisfactory. Element quality statistics and edge length distribution graphs compare the new scheme with other triangular mesh generation algorithms.

The purpose of this paper is to consider double‐diffusive convection in a heated porous medium saturated with a fluid. Of particular interest is the case where the fluid has a stabilizing concentration gradient and small diffusivity.

A fully‐coupled stabilized finite element scheme and adaptive mesh refinement (AMR) methodology are introduced to solve the resulting coupled multiphysics application and resolve fine scale solution features. The code is written on top of the open source finite element library LibMesh, and is suitable for parallel, high‐performance simulations of large‐scale problems.

The stabilized adaptive finite element scheme is used to compute steady and unsteady onset of convection in a generalized Horton‐Rogers‐Lapwood problem in both two and three‐dimensional domains. A detailed study confirming the applicability of AMR in obtaining the predicted dependence of solutal Nusselt number on Lewis number is given. A semi‐permeable barrier version of the generalized HRL problem is also studied and is believed to present an interesting benchmark for AMR codes owing to the different boundary and internal layers present in the problem. Finally, some representative adaptive results in a complex 3D heated‐pipe geometry are presented.

This work demonstrates the feasibility of stabilized, adaptive finite element schemes for computing simple double‐diffusive flow models, and it represents an easily‐generalizable starting point for more complex calculations since it is based on a highly‐general finite element library. The complementary nature of h‐adaptivity and stabilized finite element techniques for this class of problem is demonstrated using particularly simple error indicators and stabilization parameters. Finally, an interesting double‐diffusive convection benchmark problem having a semi‐permeable barrier is suggested.

The real challenge in the solution of contact problems is the lack of an optimal adaptive scheme. As the contact zone is a priori unknown, successive refinement and iterative method are necessary to obtain a high-accuracy solution. The purpose of this paper is to provide an optimal adaptive scheme based on second-generation finite element wavelets for the solution of non-linear variational inequality of the contact problem.

To generate an elementary multi-resolution mesh, the authors used hierarchical bases (HB) composed of Lagrange finite element interpolation functions. These HB functions are customized using second-generation wavelet techniques for a fast convergence rate. At each step of the algorithm, the active set method along with mesh adaptation is used for solving the constrained minimization problem of contact case. Wavelet coefficients-based error indicators are used, and computation is focused on mesh zones with a high error indication. The authors take advantage of the wavelet transform to develop a parameter-free adaptive scheme to generate an appropriate and optimal mesh.

Adaptive wavelet Galerkin scheme (AWGS), a newly developed method for multi-scale mesh adaptivity in this work, is a combination of the second-generation wavelet transform and finite element method and significantly improves the accuracy of the results without approximating an additional problem of error estimation equations. A comparative study is performed taking a solution on a highly refined mesh and results are generated using AWGS.

The proposed adaptive technique can be utilized in the simulation of mechanical and biomechanical structures where multiple bodies come into contact with each other. The algorithm of the method is easy to implement and found to be successful in producing a sufficiently accurate solution with relatively less number of mesh nodes.

Although many error estimation techniques have been developed over the past several years to solve contact problems adaptively, because of boundary non-linearity development, a reliable error estimator needs further investigation. The present study attempts to resolve this problem without having to recompute the entire solution on a new mesh.

In 1982, Smith and Hutton published comparative results of several different convection‐diffusion schemes applied to a specially devised test problem involving near‐discontinuities and strong streamline curvature. First‐order methods showed significant artificial diffusion, whereas higher‐order methods gave less smearing but had a tendency to overshoot and oscillate. Perhaps because unphysical oscillations are more obvious than unphysical smearing, the intervening period has seen a rise in popularity of low‐order artificially diffusive schemes, especially in the numerical heat‐transfer industry. This paper presents an alternative strategy of using non‐artificially diffusive higher‐order methods, while maintaining strictly monotonic transitions through the use of simple flux‐limiter constraints. Limited third‐order upwinding is usually found to be the most cost‐effective basic convection scheme. Tighter resolution of discontinuities can be obtained at little additional cost by using automatic adaptive stencil expansion to higher order in local regions, as needed.

High resolution models of air pollution transport and transformation are necessary in order to test possible abatement strategies based on pollution control and to forecast high pollution episodes. Models are especially relevant for secondary pollutants like ozone and nitrogen dioxide, which are formed in the atmosphere through nonlinear chemical reactions involving primary pollutant species, often far from their sources. Often we are trying to resolve the interactions between plumes from point sources such as power stations and regional pollution tides of ozone formed in other European countries. One method of tackling this problem of different scales is to use different grid sizes, using highly resolved grids in regions where the structure is very fine. Telescopic gridding is currently used in high emission areas or around sensitive receptor points. However, since meteorological conditions vary, this method cannot resolve a priori highly structured regions away from sources, e.g. along plumes. Such refinement can be achieved using adaptive methods which increase resolutions in regions of steep spatial gradients. This article describes the use of 3D adaptive gridding models for pollution transport and reactions using both a layered and a fully adaptive 3D tetrahedral approach and provides examples which show the effect of grid resolution on secondary pollutant formation.

The purpose of this paper is to extend the generalized finite‐difference calculus of flexible local approximation methods (FLAME) to problems where local analytical solutions are unavailable.

FLAME uses accurate local approximations of the solution to generate difference schemes with small consistency errors. When local analytical approximations are too complicated, semi‐analytical or numerical ones can be used instead. In the paper, this strategy is applied to electrostatic multi‐particle simulations and to electromagnetic wave propagation and scattering. The FLAME basis is constructed by solving small local finite‐element problems or, alternatively, by a local multipole‐multicenter expansion. As yet another alternative, adaptive FLAME is applied to problems of wave propagation in electromagnetic (photonic) crystals.

Numerical examples demonstrate the high rate of convergence of new five‐ and nine‐point schemes in 2D and seven‐ and 19‐point schemes in 3D. The accuracy of FLAME is much higher than that of the standard FD scheme. This paves the way for solving problems with a large number of particles on relatively coarse grids. FLAME with numerical bases has particular advantages for the multi‐particle model of a random or quasi‐random medium.

Irregular stencils produced by local refinement may adversely affect the accuracy. This drawback could be rectified by least squares FLAME, where the number of stencil nodes can be much greater than the number of basis functions, making the method more robust and less sensitive to the irregularities of the stencils.

Previous applications of FLAME were limited to purely analytical basis functions. The present paper shows that numerical bases can be successfully used in FLAME when analytical ones are not available.

This paper gives a bibliographical review of the finite element and boundary element parallel processing techniques from the theoretical and application points of view. Topics include: theory – domain decomposition/partitioning, load balancing, parallel solvers/algorithms, parallel mesh generation, adaptive methods, and visualization/graphics; applications – structural mechanics problems, dynamic problems, material/geometrical non‐linear problems, contact problems, fracture mechanics, field problems, coupled problems, sensitivity and optimization, and other problems; hardware and software environments – hardware environments, programming techniques, and software development and presentations. The bibliography at the end of this paper contains 850 references to papers, conference proceedings and theses/dissertations dealing with presented subjects that were published between 1996 and 2002.