Enforcement Of Dirichlet Boundary Conditions Research Articles

Direct immersogeometric fluid flow and heat transfer analysis of objects represented by point clouds

Immersogeometric analysis (IMGA) is a geometrically flexible method that enables one to perform multiphysics analysis directly using complex computer-aided design (CAD) models. While the IMGA approach is well-studied and has a remarkable advantage over traditional CFD, IMGA still requires a well-defined B-rep model to represent the geometry. Obtaining such a model can sometimes be equally as challenging as creating a body-fitted mesh. To address this issue, we develop a novel IMGA approach for the simulation of incompressible and compressible flows around complex geometries represented by point clouds in this work. The point cloud representation of geometries is a direct method for digitally acquiring geometric information using LiDAR scanners, optical scanners, or other passive methods such as multi-view stereo images. The point cloud object’s geometry is represented using a set of unstructured points in the Euclidean space with (possible) orientation information in the form of surface normals. Due to the absence of topological information in the point cloud model, there are no guarantees for the geometric representation to be watertight or 2-manifold or to have consistent normals. To perform IMGA directly using point cloud geometries, we first develop a method for estimating the inside–outside information and the surface normals directly from the point cloud. We also propose a method to compute the Jacobian determinant for the surface integration (over the point cloud) necessary for the weak enforcement of Dirichlet boundary conditions. We validate these geometric estimation methods by comparing the geometric quantities computed from the point cloud with those obtained from analytical geometry and tessellated CAD models. In this work, we also develop thermal IMGA to simulate heat transfer in the presence of flow over complex geometries. The proposed framework is tested for a wide range of Reynolds and Mach numbers on benchmark problems of geometries represented by point clouds, showing the robustness and accuracy of the method. Finally, we demonstrate the applicability of our approach by performing IMGA on large industrial-scale construction machinery represented using a point cloud of more than 12 million points.

Computational investigation of natural ventilation induced by solar chimneys: Significance of building space on thermofluid behavior

A solar chimney is a typical device to harness naturally available energy resources to power ventilation inside buildings. A comparative computational fluid dynamics study of the flow and thermal mechanics inside solar chimneys is performed in this study. We utilize a variational multiscale formulation to model the combined turbulent/laminar flow regimes presented in the natural ventilation problem in the sense of large eddy simulation. Nitsche type weak enforcement of Dirichlet boundary conditions is integrated into the numerical framework to address the excessive mesh resolution requirement in flow and thermal boundary layers. Numerical methodology is verified and validated against experimental data in a model room with a solar chimney, and good agreement between the present results and the reference data is obtained. Finally, the thermofluid characteristics are investigated in a building equipped with different solar chimney designs. Particular emphases are placed on the effects of attached building spaces to the flows within the chimneys. The results indicated that the complex and realistic building space in this paper reduces the turbulence entering the solar chimney inlet and, thus, increases the air flow rate by up to 48.9% compared with the frequently used standalone chimney simulation paradigm. In addition, the thermal comfort indices are presented. With increased air flow rate in the chimney, the overall thermal comfort in the building is likely to be decreased, suggesting the necessity in the future studies to consider thermal comfort as an optimization objective.

Variational Multiscale immersed boundary method for incompressible turbulent flows

This paper presents an immersed boundary method for weak enforcement of Dirichlet boundary conditions on surfaces that are immersed in the stationary background discretizations. An interface stabilized form is developed by applying the Variational Multiscale Discontinuous Galerkin (VMDG) method at the immersed boundaries. The formulation is augmented with a variationally derived ghost-penalty type term. The weak form of the momentum balance equations is embedded with a residual-based turbulence model for incompressible turbulent flows. A significant contribution in this work is the variationally derived analytical expression for the Lagrange multiplier for weak enforcement of the Dirichlet boundary conditions at the immersed boundary. In addition, the analytical expression for the interfacial stabilization tensor emerges which accounts for the geometric aspects of the cut elements that are produced when the immersed surface geometry traverses the underlying mesh. A unique attribute of the fine-scale variational equation is that it also yields a posteriori error estimator that can evaluate the local error in weak enforcement of the essential boundary conditions at the embedded boundaries. The method is shown to work with meshes comprised of hexahedral and tetrahedral elements. Numerical experiments show that the norm of the stabilization tensor varies spatially and temporally as a function of the flow physics at the embedded boundary. Test cases with increasing levels of complexity are presented to validate the method on benchmark problems of flows around cylindrical and spherical geometric shapes, and turbulent features of the flows are analyzed.

Enforcing essential boundary conditions on domains defined by point clouds

This paper develops and investigates a new method for the application of Dirichlet boundary conditions for computational models defined by point clouds. Point cloud models often stem from laser or structured-light scanners which are used to scan existing mechanical structures for which CAD models either do not exist or from which the artifact under investigation deviates in shape or topology. Instead of reconstructing a CAD model from point clouds via surface reconstruction and a subsequent boundary conforming mesh generation, a direct analysis without pre-processing is possible using embedded domain finite element methods. These methods use non-boundary conforming meshes which calls for a weak enforcement of Dirichlet boundary conditions. For point cloud based models, Dirichlet boundary conditions are usually imposed using a diffuse interface approach. This leads to a significant computational overhead due to the necessary computation of domain integrals. Additionally, undesired side effects on the gradients of the solution arise which can only be controlled to some extent. This paper develops a new sharp interface approach for point cloud based models which avoids both issues. The computation of domain integrals is circumvented by an implicit approximation of corresponding Voronoi diagrams of higher order and the resulting sharp approximation avoids the side effects of diffuse approaches. Benchmark examples from the graphics as well as the computational mechanics community are used to verify the algorithm. All algorithms are implemented in the FCMLab framework and provided at https://gitlab.lrz.de/cie_sam_public/fcmlab/. Further, we discuss challenges and limitations of point cloud based analysis w.r.t. application of Dirichlet boundary conditions.

The finite cell method with least squares stabilized Nitsche boundary conditions

We apply the recently developed least squares stabilized symmetric Nitsche method for enforcement of Dirichlet boundary conditions to the finite cell method. The least squares stabilized Nitsche method in combination with finite cell stabilization leads to a symmetric positive definite stiffness matrix and relies only on elementwise stabilization, which does not lead to additional fill in. We prove a priori error estimates and bounds on the condition numbers.

Weakly imposed boundary conditions for shear-rate dependent non-Newtonian fluids: application to cardiovascular flows.

This paper presents a stabilized formulation for the generalized Navier-Stokes equations for weak enforcement of essential boundary conditions. The non-Newtonian behavior of blood is modeled via shear-rate dependent constitutive equations. The boundary terms for weak enforcement of Dirichlet boundary conditions are derived via locally resolving the fine-scale variational equation facilitated by the Variational Multiscale (VMS) framework. The proposed method reproduces the consistency and stabilization terms that are present in the Nitsche type approaches. In addition, for the shear-rate fluids, two more boundary terms appear. One of these terms is the viscosity-derivative term and is a function of the shear-rate, while the other term is a zeroth-order term. These terms play an important role in attaining optimal convergence rates for the velocity and pressure fields in the norms considered. A most significant contribution is the form of the stabilization tensors that are also variationally derived. Employing edge functions the edge stabilization tensor is numerically evaluated, and it adaptively adjusts itself to the magnitude of the boundary residual. The resulting formulation is variationally consistent and the weakly imposed no-slip boundary condition leads to higher accuracy of the spatial gradients for coarse boundary-layer meshes when compared with the traditional strongly imposed boundary conditions. This feature of the present approach will be of significance in imposing interfacial continuity conditions across non-matching discretizations in blood-artery interaction problems. A set of test cases is presented to investigate the mathematical attributes of the method and a patient-specific case is presented to show its clinical relevance.

Nitsche’s method for linear Kirchhoff–Love shells: Formulation, error analysis, and verification

Stable and accurate modeling of thin shells requires proper enforcement of all types of boundary conditions. Unfortunately, for Kirchhoff–Love shells, strong enforcement of Dirichlet boundary conditions is difficult because both displacement and normal rotation boundary conditions must be applied. A popular alternative is to employ Nitsche’s method to weakly enforce all boundary conditions. However, while many Nitsche-based formulations have been proposed in the literature, they lack comprehensive error analyses and verification. In fact, existing formulations are variationally inconsistent and yield sub-optimal convergence rates when used with common boundary condition specifications. In this paper, we present a novel Nitsche-based formulation for the linear Kirchhoff–Love shell that is provably stable and optimally convergent for general sets of admissible boundary conditions. To arrive at our formulation, we first present a framework for constructing Nitsche’s method for any abstract variationally constrained minimization problem. We then apply this framework to the linear Kirchhoff–Love shell and, for the particular case of NURBS-based isogeometric analysis, we prove that the resulting formulation yields optimal convergence rates in both the shell energy norm and the standard L2-norm. To arrive at this formulation, we derive the Euler–Lagrange equations for general sets of admissible boundary conditions and show that the Euler–Lagrange boundary conditions typically presented in the literature are incorrect. We verify our formulation by manufacturing solutions for a new shell obstacle course that encompasses flat, parabolic, hyperbolic, and elliptic geometric configurations with a variety of common boundary condition specifications. These manufactured solutions allow us to robustly measure the error across the entire shell in contrast with current best practices where displacement and stress errors are only measured at specific locations. We use NURBS discretizations to represent the shell geometry and show optimal convergence rates in both the shell energy norm and the standard L2-norm with varying polynomial degrees for all of the problems in the obstacle course.

A Variational Multiscale method with immersed boundary conditions for incompressible flows.

This paper presents a new stabilized form of incompressible Navier-Stokes equations for weak enforcement of Dirichlet boundary conditions at immersed boundaries. The boundary terms are derived via the Variational Multiscale (VMS) method which involves solving the fine-scale variational problem locally within a narrow band along the boundary. The fine-scale model is then variationally embedded into the coarse-scale form that yields a stabilized method which is free of user defined parameters. The derived boundary terms weakly enforce the Dirichlet boundary conditions along the immersed boundaries that may not align with the inter-element edges in the mesh. A unique feature of this rigorous derivation is that the structure of the stabilization tensor which emerges is naturally endowed with the mathematical attributes of area-averaging and stress-averaging. The method is implemented using 4-node quadrilateral and 8-node hexahedral elements. A set of 2D and 3D benchmark problems is presented that investigate the mathematical attributes of the method. These test cases show that the proposed method is mathematically robust as well as computationally stable and accurate for modeling boundary layers around immersed objects in the fluid domain.

An implicit boundary finite element method with extension to frictional sliding boundary conditions and elasto-plastic analyses

Implicit boundary methods, which enrich the interpolation structure with implicit weight functions, are straightforward methods for the enforcement of Dirichlet boundary conditions. In this article, we follow the implicit boundary method that uses approximate step functions (the step boundary method) developed by Kumar et al. and provide modifications that have several advantages. Roller boundary conditions have wide practical applications in engineering, however, the step boundary method for roller boundary conditions with inclinations has yet to be fully formulated through to the final linear system of equations. Thus we provide a complete derivation that leads to simplified stiffness matrices compared to the original approach, which can be implemented directly in fictitious domain finite element analysis. The approach is then extended, we believe for the first time, to the nonlinear cases of frictional boundary conditions and elasto-plastic material behaviour. The proposed formulation and procedures are validated on a number of example problems that test different aspects of the method.

A residual-based variational multiscale method with weak imposition of boundary conditions for buoyancy-driven flows

There is growing interest in high fidelity simulations of buoyancy-driven flows in indoor environments. This is driven to a large extent by the push to utilize naturally occurring flows to reduce energy consumption in the built environment. Naturally occurring (buoyancy-driven) flows exhibit spatiotemporal thermal fluctuations, and accurately capturing these thermal variations is essential to calculate energy usage. In this work, we deploy a residual-based variational multiscale (VMS) finite element large-eddy simulation model for accurately modeling buoyancy-driven flows in enclosed environments. We use the canonical example of the Rayleigh–Bénard convection problem in 2D and 3D to verify and validate the VMS model. We show good comparison with benchmark numerical and experimental results across seven orders of magnitude variation in Rayleigh numbers (Ra∼103 to 1010), covering laminar, transition, and turbulent regimes without any extra treatments. We additionally employ weak enforcement of Dirichlet boundary conditions for both velocity and temperature, and show that comparable results can be produced with much coarser meshes. This confirms that the VMS framework with weak imposition of Dirichlet boundary conditions is a computationally efficient approach to model buoyancy-driven flow physics in complex indoor environments.

A new least squares stabilized Nitsche method for cut isogeometric analysis

We derive a new stabilized symmetric Nitsche method for enforcement of Dirichlet boundary conditions for elliptic problems of second order in cut isogeometric analysis (CutIGA). We consider C1 splines and stabilize the standard Nitsche method by adding a certain elementwise least squares terms in the vicinity of the Dirichlet boundary and an additional term on the boundary which involves the tangential gradient. We show coercivity with respect to the energy norm for functions in H2(Ω) and optimal order a priori error estimates in the energy and L2 norms. To obtain a well posed linear system of equations we combine our formulation with basis function removal which essentially eliminates basis functions with sufficiently small intersection with Ω. The upshot of the formulation is that only elementwise stabilization is added in contrast to standard procedures based on ghost penalty and related techniques and that the stabilization is consistent. In our numerical experiments we see that the method works remarkably well in even extreme cut situations using a Nitsche parameter of moderate size.

Computer Modeling of Wind Turbines: 1. ALE-VMS and ST-VMS Aerodynamic and FSI Analysis

This is the first part of a two-part article on computer modeling of wind turbines. We describe the recent advances made by our teams in ALE-VMS and ST-VMS computational aerodynamic and fluid–structure interaction (FSI) analysis of wind turbines. The ALE-VMS method is the variational multiscale version of the Arbitrary Lagrangian–Eulerian method. The VMS components are from the residual-based VMS method. The ST-VMS method is the VMS version of the Deforming-Spatial-Domain/Stabilized Space–Time method. The ALE-VMS and ST-VMS serve as the core methods in the computations. They are complemented by special methods that include the ALE-VMS versions for stratified flows, sliding interfaces and weak enforcement of Dirichlet boundary conditions, ST Slip Interface (ST-SI) method, NURBS-based isogeometric analysis, ST/NURBS Mesh Update Method (STNMUM), Kirchhoff–Love shell modeling of wind-turbine structures, and full FSI coupling. The VMS feature of the ALE-VMS and ST-VMS addresses the computational challenges associated with the multiscale nature of the unsteady flow, and the moving-mesh feature of the ALE and ST frameworks enables high-resolution computation near the rotor surface. The ST framework, in a general context, provides higher-order accuracy. The ALE-VMS version for sliding interfaces and the ST-SI enable moving-mesh computation of the spinning rotor. The mesh covering the rotor spins with it, and the sliding interface or the SI between the spinning mesh and the rest of the mesh accurately connects the two sides of the solution. The ST-SI also enables prescribing the fluid velocity at the turbine rotor surface as weakly-enforced Dirichlet boundary condition. The STNMUM enables exact representation of the mesh rotation. The analysis cases reported include both the horizontal-axis and vertical-axis wind turbines, stratified and unstratified flows, standalone wind turbines, wind turbines with tower or support columns, aerodynamic interaction between two wind turbines, and the FSI between the aerodynamics and structural dynamics of wind turbines. Comparisons with experimental data are also included where applicable. The reported cases demonstrate the effectiveness of the ALE-VMS and ST-VMS computational analysis in wind-turbine aerodynamics and FSI.

Direct immersogeometric fluid flow analysis using B-rep CAD models

We present a new method for immersogeometric fluid flow analysis that directly uses the CAD boundary representation (B-rep) of a complex object and immerses it into a locally refined, non-boundary-fitted discretization of the fluid domain. The motivating applications include analyzing the flow over complex geometries, such as moving vehicles, where the detailed geometric features usually require time-consuming, labor-intensive geometry cleanup or mesh manipulation for generating the surrounding boundary-fitted fluid mesh. The proposed method avoids the challenges associated with such procedures. A new method to perform point membership classification of the background mesh quadrature points is also proposed. To faithfully capture the geometry in intersected elements, we implement an adaptive quadrature rule based on the recursive splitting of elements. Dirichlet boundary conditions in intersected elements are enforced weakly in the sense of Nitsche's method. To assess the accuracy of the proposed method, we perform computations of the benchmark problem of flow over a sphere represented using B-rep. Quantities of interest such as drag coefficient are in good agreement with reference values reported in the literature. The results show that the density and distribution of the surface quadrature points are crucial for the weak enforcement of Dirichlet boundary conditions and for obtaining accurate flow solutions. Also, with sufficient levels of surface quadrature element refinement, the quadrature error near the trim curves becomes insignificant. Finally, we demonstrate the effectiveness of our immersogeometric method for high-fidelity industrial scale simulations by performing an aerodynamic analysis of an agricultural tractor directly represented using B-rep.

The tetrahedral finite cell method: Higher‐order immersogeometric analysis on adaptive non‐boundary‐fitted meshes

SummaryThe finite cell method (FCM) is an immersed domain finite element method that combines higher‐order non‐boundary‐fitted meshes, weak enforcement of Dirichlet boundary conditions, and adaptive quadrature based on recursive subdivision. Because of its ability to improve the geometric resolution of intersected elements, it can be characterized as an immersogeometric method. In this paper, we extend the FCM, so far only used with Cartesian hexahedral elements, to higher‐order non‐boundary‐fitted tetrahedral meshes, based on a reformulation of the octree‐based subdivision algorithm for tetrahedral elements. We show that the resulting TetFCM scheme is fully accurate in an immersogeometric sense, that is, the solution fields achieve optimal and exponential rates of convergence for h‐refinement and p‐refinement, if the immersed geometry is resolved with sufficient accuracy. TetFCM can leverage the natural ability of tetrahedral elements for local mesh refinement in three dimensions. Its suitability for problems with sharp gradients and highly localized features is illustrated by the immersogeometric phase‐field fracture analysis of a human femur bone. Copyright © 2016 John Wiley & Sons, Ltd.

A mixed/hybrid Dirichlet formulation for wall-bounded flow problems including turbulent flow

In this study, a new method for the weak imposition of Dirichlet boundary conditions in convection-dominated convection–diffusion and wall-bounded turbulent flow problems is developed. It is based on an embedded Dirichlet formulation which utilises an additional discontinuous stress field for imposing boundary conditions. Values at outflow boundaries may not be sufficiently controllable on underresolved meshes, e.g., when using Nitsche’s method for weak enforcement of Dirichlet boundary conditions. Hence, a consistent modification of an earlier-proposed constant-parameter embedded Dirichlet formulation that allows to regain control if required is introduced. A separate treatment of boundary conditions in wall-normal and tangential direction is enabled. This split is physically motivated, and it turns out to be crucial for a successful application of the method. On the one hand, it allows for including a formulation based on the law of the wall in tangential direction. On the other hand, a consistent modification derived from theoretical considerations for convection–diffusion problems is applied in wall-normal direction, preventing unphysical penetration of walls. Numerical results obtained for turbulent channel flow at modest friction Reynolds number Reτ=395 employing meshes with and without adequate resolution of the boundary layer are presented. Superior performance of the proposed method compared to strongly-imposed boundary conditions is demonstrated by an error reduction of up to 50% for mean velocity results in the core of the channel on coarse discretisations.

A NORMED G SPACE AND WEAKENED WEAK (W2) FORMULATION OF A CELL-BASED SMOOTHED POINT INTERPOLATION METHOD

This paper presents a normed G1 space and a weakened weak (W2) formulation of a cell-based smoothed point interpolation method (CS-PIM) for 2D solid mechanics problems using three-node triangular cells. Displacement fields in the CS-PIM are constructed using the point interpolation method (polynomial PIM or radial PIM) and hence the shape functions possess the Kronecker delta property facilitating the easy enforcement of Dirichlet boundary conditions. The edge-based T-schemes are introduced for selecting supporting nodes for creating the PIM shape functions and an adaptive coordinate transformation (CT) technique is proposed to solve the singularity problem for the moment matrix. Smoothed strains are obtained by performing the generalized smoothing operation over each triangular background cell. Because the nodal PIM shape functions can be discontinuous, a W2 formulation of generalized smoothed Galerkin (GS-Galerkin) weak form is then used to create the discretized system equations. Numerical examples including static, free and forced vibration problems have been studied to examine the present method in terms of accuracy, convergence, efficiency and temporal stability.

EDGE-BASED SMOOTHED POINT INTERPOLATION METHODS

This paper formulates an edge-based smoothed point interpolation method (ES-PIM) for solid mechanics using three-node triangular meshes. In the ES-PIM, displacement fields are construed using the point interpolation method (polynomial PIM or radial PIM), and hence the shape functions possess the Kronecker delta property, facilitates the enforcement of Dirichlet boundary conditions. Strains are obtained through smoothing operation over each smoothing domain associated with edges of the triangular background cells. The generalized smoothed Galerkin weak form is then used to create the discretized system equations and the formation is weakened weak formulation. Four schemes of selecting nodes for interpolation using the PIM have been introduced in detail and ES-PIM models using these four schemes have been developed. Numerical studies have demonstrated that the ES-PIM possesses the following good properties: (1) the ES-PIM models have a close-to-exact stiffness, which is much softer than for the overly-stiff FEM model and much stiffer than for the overly-soft node-based smoothed point interpolation method (NS-PIM) model; (2) results of ES-PIMs are generally of superconvergence and "ultra-accurate"; (3) no additional degrees of freedom are introduced, the implementation of the method is straightforward, and the method can achieve much better efficiency than the FEM using the same set of triangular meshes.

Enforcement of boundary conditions in meshfree methods using interpolating moving least squares

A technique for enforcement of boundary conditions in meshfree methods based on interpolating moving least squares (IMLS) via usage of singular weights for the solution of partial differential equations is presented. Due to the specific singular choice of the weight functions, which is needed to guarantee the interpolation, there is a problem to find the inverse of the singular matrix. We extend the perturbation technique applied to transport equations by Kunle [Entwicklung und Untersuchung von Moving Least Square Verfahren zur numerischen Simulation hydrodynamischer Gleichungen, Dissertation, Fakultät für Physik, Eberhard-Karls-Universität zu Tübingen] to allow the correct evaluation of all necessary derivatives for the meshfree collocation solution of boundary value problems in interpolation points at a reasonable cost. The inverse is carried out using the regularized weight function. It turns out that a stable inverse is obtained when the vanishing regularization parameter is considered. The detailed description of this strategy is presented. In contrast to standard kernel functions used in EFGM, RKPM, etc., the singular kernel functions lead to truly interpolating functions which satisfy the Kronecker-delta property and they can therefore be used for enforcement of Dirichlet boundary conditions when solving boundary value problems. As a confirmation of the mathematical derivations we give a solution to a model BVP with a known analytical solution, as well as the experimental convergence study of the method to analytical solutions.

Meshfree collocation solution of boundary value problems via interpolating moving least squares

AbstractThis paper presents a meshfree interpolating moving least squares (IMLS) method based on singular weights for the solution of partial differential equations. Due to the specific singular choice of weight functions, which is needed to guarantee the interpolation, there arises a problem of finding the inverse of the occurring singular matrix. The inverse is carried out using a regularization of weight functions. It turns out that a stable inverse is obtained by considering the vanishing regularization parameter.Moreover, the use of this perturbation technique allows the correct evaluation of all necessary derivatives in interpolation points at a reasonable cost.Unlike standard kernel functions used in EFGM, RKPM, etc., the singular kernel functions lead to really interpolating functions which satisfy the Kronecker‐delta property. They can be used for enforcement of Dirichlet boundary conditions when solving boundary value problems.Solution to a model BVP as well as an experimental convergence study of the method to analytical solutions, confirming the mathematical derivations, are given. Copyright © 2006 John Wiley & Sons, Ltd.

On 2D elliptic discontinuous Galerkin methods

AbstractWe discuss the discretization using discontinuous Galerkin (DG) formulation of an elliptic Poisson problem. Two commonly used DG schemes are investigated: the original average flux proposed by Bassi and Rebay (J. Comput. Phys. 1997; 131:267) and the local discontinuous Galerkin (LDG) (SIAM J. Numer. Anal. 1998; 35:2440–2463) scheme. In this paper we expand on previous expositions (Discontinuous Galerkin Methods: Theory, Computation and Applications. Springer: Berlin, 2000; 135–146; SIAM J. Sci. Comput. 2002; 24(2):524–547; Int. J. Numer. Meth. Engng. 2003; 58(2): 1119–1148) by adopting a matrix based notation with a view to highlighting the steps required in a numerical implementation of the DG method. Through consideration of standard C0‐type expansion bases, as opposed to elementally orthogonal expansions, with the matrix formulation we are able to apply static condensation techniques to improve efficiency of the direct solver when high order expansions are adopted. The use of C0‐type expansions also permits the direct enforcement of Dirichlet boundary conditions through a ‘lifting’ approach where the LDG flux does not require further stabilization. In our construction we also adopt a formulation of the continuous DG fluxes that permits a more general interpretation of their numerical implementation. In particular it allows us to determine the conditions under which the LDG method provides a near local stencil. Finally a study of the conditioning and the size of the null space of the matrix systems resulting from the DG discretization of the elliptic problem is undertaken. Copyright © 2005 John Wiley & Sons, Ltd.