The workshop schedule appears below. All talks will be held in the Student Center Theater on the 1st (ground) floor of the Georgia Tech Student Center, which is located at 350 Ferst Drive, Atlanta, GA 30313, a short walk from the Hampton Inn Atlanta-Georgia Tech. The poster session on Tuesday will be held in a different location, as indicated below.

Mon Oct 26    
0800-0850 Registration  
0850-0900 Welcoming remarks  
0900-1000 Franco Brezzi (plenary) Serendipity Virtual Element Methods
1000-1030 coffee  
1030-1055 Alessandro Russo Virtual Element Method for general second order elliptic problems
1055-1120 Marco Verani A virtual element method for the Cahn-Hilliard problem
1120-1150 Stefano Berrone The Virtual Element Method for Discrete Fracture Network flow simulations
1150-1200 group photo - all participants Please join us for a quick group photo on the stairs of Georgia Tech plaza before breaking for lunch.
1200-1315 lunch  
1315-1415 Junping Wang (plenary) Basic Principles of Weak Galerkin Finite Element Methods for PDEs
1415-1445 coffee  
1445-1510 Xiu Ye New class of finite element methods: Weak Galerkin Methods
1510-1535 Ran Zhang The weak Galerkin method for Laplacian eigenvalue problems
1535-1600 Chunmei Wang A Primal-Dual Weak Galerkin Finite Element Method for Second Order Elliptic Equations in Nondivergence Form
1800- ... dinner Please join us for dinner and/or drinks at the upstairs room of Publik Draft House, located at 654 Peachtree St NE, Atlanta, GA 30308, starting at 6pm. Please note that you will have to pay your own bill.
Tue Oct 27    
0800-0900 Registration  
0900-1000 Glaucio Paulino (plenary) Why Polygons?
1000-1030 coffee  
1030-1055 Eugene Wachspress Rational Barycentrics: Polygons and Polycons
1055-1120 Ming-Jun Lai Polygonal Splines and Their Applications for Numerical Solution of PDE
1120-1150 Yanqiu Wang Anisotropic quality measures and adaptation for polygonal meshes
1150-1315 lunch  
1315-1340 Long Chen An interface-fitted mesh generator and finite element methods for elliptic interface problems in two and three dimensions
1340-1405 Yingjie Liu DG with conservative constraints for improving the CFL numbers
1405-1430 Jiangguo Liu C++ Polymorphism for Weak Galerkin Finite Element Methods on Polytopal Meshes
1430-1500 coffee  
1500-1525 Zhaonan Dong hp-Version discontinuous Galerkin methods for advection-diffusion-reaction problems on polytopic meshes
1525-1550 Paola Antonietti Multigrid algorithms for hp-version Discontinuous Galerkin methods on polytopic meshes
1630-1730 Poster Session The poster session will be held in the Skiles classroom building ground floor hallway and room 006. Here is a list of poster presenters and titles
Wed Oct 28    
0900-1000 Konstantin Lipnikov (plenary) Design principles of the mimetic finite difference schemes
1000-1030 coffee  
1030-1055 Simon Lemaire A review of Hybrid High-Order methods: formulations, computational aspects, links with other methods
1055-1120 Scott Mitchell VoroCrust: Simultaneous Surface Reconstruction and Volume Meshing with Voronoi cells
1120-1150 Lin Mu Weak Galerkin Finite Element Methods and Numerical Applications
1150-1315 lunch  
1315-1340 Yann Savoye "Free at Last": Cage-based Living Geometry
1340-1405 N. Sukumar Numerical integration of homogeneous functions on convex and nonconvex polygons and polyhedra
1405-1430 Xiaozhe Hu Finite Element Multigrid Framework for Mimetic Finite Difference Discretizations
1430-1445 closing remarks /discussion

Poster session information:

A poster session will be held in Room 006 of the Skiles classroom building from 4:30-5:30pm on Tuesday October 27.

Poster presenters should bring their posters to the conference on Tuesday morning; posters will be mounted by Georgia Tech staff during the lectures on Tuesday. Additional details will be provided during the conference on Monday.

Poster presenters, co-authors, and titles

Omar Al-Hinai (University of Texas at Austin)
with: Mary Wheeler, Ivan Yotov
"A Generalized Mimetic Finite Difference Method over Voronoi Diagrams"

Michele Botti (University of Montpellier)
with: Daniele Boffi, Daniele Antonio Di Pietro
"A nonconforming high-order method for the Biot problem",

Jesse Chan (Virginia Tech)
with: Zheng Wang, Axel Modave, J.F. Remacle, T. Warburton
"Efficient DG methods on hybrid meshes"

Florent Chave (University of Montpellier)
with: Daniele Antonio Di Pietro, Fabien Marche, Franck Pigeonneau
"A Hybrid High Order method for the Cahn-Hilliard equation"

Eric B. Chin (University of California, Davis)
with: Jean B. Lasserre, N. Sukumar
"Numerical integration of homogeneous functions and polynomials over polytopes"

Federico Fuentes (University of Texas at Austin)
with: Brendan Keith and Leszek Demkowicz
"Pyramid High Order Exact Sequence Shape Functions"

Brendan Keith (University of Texas at Austin)
with: F. Fuentes, L. Demkowicz
"Orientation Embedded High Order Shape Functions for the Exact Sequence Elements of All Shapes"

Liangwei Li (The George Washington University)
with: Zhen Zhang, Lin Mu,  Chunlei Liang, Junping Wang, James Lee
"A Parallel Implicit Scheme for Geometric Nonlinear Elastic Equation Based on Hybrid Weak Galerkin - Continuous Galerkin Element"

Oliver Sutton (University of Leicester)
with: Andrea Cangiani, Manolis Georgoulis, Tristan Pryer
"Residual a posteriori error estimation for virtual element methods"

Giuseppe Vacca (University of Bari)

with: Lourenco Beirao da Veiga, Carlo Lovadina
"Divergence free Virtual Elements for the Stokes problem"

Chi-Jen Wang (Georgia Institute of Technology)
"Lattice Differential Equation analysis of Schloegl's second model for particle creation and annihilation"

Chunmei Wang (Georgia Institute of Technology)
with: Junping Wang
"Weak Galerkin Methods for PDEs"

Steven Wopschall (University of California, Davis)
with: Brian D. Giffin, Mark M. Rashid
"The Partitioned Element Method on Hexahedral Meshes"

Tuo Zhao (Georgia Institute of Technology)
with: Adeildo S. Ramos Jr., Glaucio H. Paulino
"Topology Optimization with Nonlinear Constitutive Model Governed by von Mises Criteria"

Abstracts of talks:

Franco Brezzi: Serendipity Virtual Element Methods

After recalling the definition of Virtual Element Methods (VEMs) in their original formulation, I will introduce a new version of Virtual Elements that preserves all the good properties of the original ones, but allows a reduced number of degrees of freedom.

In particular, we will see that the Serendipity VEMs, on simplexes, have the same number of degrees of freedom as the classical Lagrange Finite Element Methods of the same degree of accuracy. As the number of edges/faces increases, the number of necessary internal degrees of freedom (for the same given accuracy) decreases. On quadrilaterals, the new VEMs remind the Serendipity Finite Elements (whence the name): the two methods (S-VEM and S-FEM) are different, but (always on quadrilaterals) they have the same number of degrees of freedom. However the S-VEM are much more insensitive to distortion.

Alessandro Russo: Virtual Element Method for elliptic problems in primal and mixed form

The Virtual Element Method (VEM in short) is a new family of high-order polygonal and polyhedral Finite Elements which have been developed quite recently. In this talk I will explain how the VEM works for general elliptic equations with variable coefficients in primal and mixed form.

Marco Verani: A virtual element method for the Cahn-Hilliard problem

Due to the wide spectrum of applications (e.g. phase separation in binary alloys, tumor growth, galaxy formation, foam formation, solidification processes and image processing) the study of efficient numerical methods for the approximate solution of Cahn-Hilliard equations has been the object of an intensive research activity.

As the Cahn-Hilliard problem is a fourth order nonlinear equation, a natural numerical approach is to resort to the use of C^1 finite elements. However, the well known difficulties related to the practical implementation of C^1 finite elements have represented so far an important obstruction that has drastically limited their use in practical applications, thus paving the road to the use of mixed methods (with an increase of the numbers of degrees of freedom, and thus of the computational cost).

In this talk we introduce and analyze a C^1 virtual element method (VEM) for the approximate solution of the Cahn-Hilliard problem. We show convergence of the approximation scheme and investigate its performance numerically.
This is a joint work with P. F. Antonietti, L. Beirao da Veiga and S. Scacchi.

Stefano Berrone: The Virtual Element Method for Discrete Fracture Network flow simulations

Subsurface fluid flow has applications in a wide range of fields, including e.g. oil/gas recovery, gas storage, pollutant percolation, water resources monitoring. Underground fluid flow in fractured media is a heterogeneous multi-scale phenomenon that involves complex geological configurations; a possible approach for modeling the phenomenon is given by Discrete Fracture Networks (DFNs), which are complex sets of polygonal intersecting fractures generated from random distributions for hydrogeological properties and geometrical features (such as orientation in the three dimensional space, position, dimension, and aspect ratio). Geological fractured media are therefore characterized by a very challenging geometrical complexity, which is one of the major difficulties to be tackled when performing flow simulations.

This talk concerns the application of the Virtual Element Method to the steady state simulation of the flow in DFNs. In this approach we can exploit the flexibility of VEM in order to tackle the geometrical complexity. Indeed, a crucial issue in DFN flow simulations is the need to provide on each fracture a good quality mesh on any randomly generated configuration. Namely, if classical triangular or quadrilateral meshes on the fractures are required to be conforming to the traces (fracture intersections), and also conforming each other, the meshing process for each fracture is not independent of the others, thus yielding in practice a quite demanding computational effort for the mesh generation process. In some cases, the meshing process may even result infeasible so that some authors propose to modify the DFN removing problematic fractures.

Here, the VEM will be used within several possible approaches to the problem: in conjunction with a newly conceived PDE-constrained optimization approach, in conjunction with a mortar approach as well as on a totally conforming polygonal mesh. Indeed, taking advantage from the great flexibility of VEM in allowing the use of rather general polygonal mesh elements, a suitable mesh for representing the solution and imposing matching conditions between the solutions on different fractures can be easily obtained, starting from an arbitrary triangular mesh independently built on each fracture, and independent of the trace disposition. Robustness and efficiency of the approach allow the application of the method to Uncertainty Quantification analysis.

Junping Wang: Basic Principles of Weak Galerkin Finite Element Methods for PDEs

The speaker shall introduce a new numerical technique, called weak Galerkin finite element method (WG), for partial differential equations. Weak Galerkin is a finite element method for PDEs where the differential operators (e.g., gradient, divergence, curl, Laplacian etc.) in the variational forms are approximated by weak forms as generalized distributions. The WG discretization procedure often involves the solution of inexpensive problems defined locally on each element. The solution from the local problems can be regarded as a reconstruction of the corresponding differential operators. The fundamental difference between the weak Galerkin finite element method and other existing methods is the use of weak functions and weak derivatives (i.e., locally reconstructed differential operators) in the design of numerical schemes based on existing variational forms for the underlying PDE problem. Weak Galerkin is a natural extension of the conforming Galerkin finite element method. Due to its great structural flexibility, the weak Galerkin finite element method is well suited to many partial differential equations by providing the needed stability and accuracy in approximation.

The talk will start with the second order elliptic equation, for which WG shall be applied and explained in detail. In particular, the concept of weak gradient will be introduced and discussed for its role in the design of weak Galerkin finite element schemes. The speaker will then introduce a general notion of weak differential operators, such as weak Hessian, weak divergence, and weak curl etc. These weak differential operators shall serve as building blocks for WG finite element methods for other classes of partial differential equations, such as the Stokes equation, the biharmonic equation, the Maxwell equations in electron magnetics theory, div-curl systems, and PDEs in non-divergence form (such as the Fokker-Planck equation). The speaker will demonstrate how WG can be applied to some model PDEs, with a discussion on their main features and advantages. Furthermore, a mathematical convergence theory shall be briefly given for some applications. The talk should be accessible to graduate students with adequate training in computational methods.

Xie Ye: Recent Developments of Weak Galerkin Methods

Employing discontinuous elements and share the simple formulations of continuous finite element methods at the same time. The Weak Galerkin method is an extension of the standard Galerkin finite element method where classical derivatives were substituted by weakly defined derivatives on functions with discontinuity. Recent development of weak Galerkin methods will be discussed in the presentation.

Chunmei Wang: A Primal-Dual Weak Galerkin Finite Element Method for Second Order Elliptic Equations in Nondivergence Form

In this talk, the speaker will introduce a discretization scheme for second order elliptic equations in nondivergence form by using weak Galerkin finite element methods. This new numerical algorithm is based on a primal-dual formulation of the second order elliptic equation in which the usual stabilizer in the weak Galerkin finite element method is employed to link the two apparently disjoint problems. An abstract framework of primal-dual weak Galerkin will be presented for problems satisfying inf-sup conditions. The speaker will outline some optimal order error estimates for the primal-dual weak Galerkin FEM, and then demonstrate the efficiency and accuracy of the method for several test problems with continuous and discontinuous coefficient matrices for which no divergence forms are possible.

Glaucio Paulino: Why polygons?

A better title for this presentation might be: why not polygons? This presentation addresses the fact that, unlike standard finite element discretizations, polygonal/polyhedral elements are suitable to accommodate very large local deformations and naturally equipped to mesh microstructures with different length scales, two essential features needed in the microscopic/mesoscopic analysis of soft solids. We discuss a computational paradigm to effectively study from the bottom-up the mechanical/physical response and stability of materials subjected to both small and finite deformations. The discretization approaches involve arbitrarily shaped elements in the context of finite element and virtual element (VEM) methods. Applications will be presented in the fields of soft materials, topology optimization, fluid mechanics and fracture mechanics.

Eugene Wachspress: Algebraic Elements

Sides of algebraic elements may be algebraic curves or surfaces of any order. Rational basis functions that achieve any degree of polynomial approximation may be constructed. The degree-one bases are generalized barycentric coordinates. One simple application is replacement of adjacent sides of concave vertices of a star-polygon by parabolas with mid-side nodes at the vertices. The irrational star-polygon barycentrics are thus replaced by rational degree-one bases. These are regular within the element.

Ming-Jun Lai: A Construction of polygonal/polyhedral splines and their application for PDE

I shall explain how to use generalized barycentric coordinates and polynomial
blossom form to construct a class of polygonal splines in 2D first. Then I will present some numerical results on using them for solution of the Poisson equation. Mainly, I shall demonstrate that the efficiency of these polygonal splines in different categories of timing. Finally, I shall explain how to extend the construction to the 3D setting for polyhedral splines. Some numerical results will be shown.

Yanqiu Wang: Anisotropic quality measures and adaptation for polygonal meshes

Three sets of anisotropic alignment and equidistribution measures are developed for polygonal meshes, one based on least squares fitting, one based on generalized barycentric mapping, and the other based on singular value decomposition of edge matrices. Based on one of these sets of quality measures and using a moving mesh method, anisotropic adaptive polygonal meshes are generated which provides nearly optimal approximation to anisotropic second order diffusion equations. Numerical examples are presented. This is a joint work with Dr. Weizhang Huang from Kansas University.

Ran Zhang: Some progress on Weak Galerkin Finite Element Scheme

The weak Galerkin (WG) finite element method is a newly developed and efficient numerical technique for solving partial differential equations (PDEs). It was first introduced and analyzed for second order elliptic equations and further applied to several other model equations, such as the biharmonic, Stokes equations to demonstrate its power and efficiency as an emerging new numerical method.
This talk introduces some new progress on WG scheme, which includes the superconvergence results on WG, the applications on eigenvalue problems, the Cahn-Hilliard equation, etc.

Long Chen: An interface-fitted mesh generator and finite element methods for elliptic interface problems in two and three dimensions

In this work, we develop a simple interface-fitted mesh algorithm which can produce an interface-fitted mesh in two and three dimension quickly. Elements in such interface-fitted mesh can be polygon or polyhedron not restricted to simplex. We thus apply virtual element methods to solve the elliptic interface problem in two and three dimensions. We present some numerical results to illustrate the effectiveness of our method.

This is a joint work with Huayi Wei and Min Wen.

Yingjie Liu: Runge-Kutta discontinuous Galerkin method with conservation constraints to improve the CFL condition

This talk is based on a recently published joint paper with my collaborators Zhiliang Xu and Xu-yan Chen. Runge-Kutta discontinuous Galerkin (RKDG) method has found many applications in fluid mechanics and many other areas since it's compact and can be formulated on unstructured meshes naturally with any order of accuracy. However it's found that its time step size decreases quickly with increasing order. We have developed a method to improve its time step size at least 3 times with essentially the same amount of work, while keeping all other nice features of the RKDG method such as its compactness. The idea is to introduce extra conservation constraints to the numerical solution. The conservation constraints are used as penalty terms to the variational energy functional of the RKDG
method, thus eliminating the need for Lagrangian multipliers.

Paola Antonietti: Multigrid algorithms for hp-version Discontinuous Galerkin methods on polytopic meshes

We present a class of multigrid algorithms for the efficient solution of the linear system of equations arising from hp-version discontinuous Galerkin discretizations of second-order elliptic problems on polytopic meshes. We prove that, under suitable assumptions on the quality of the agglomerated coarse grids, the multigrid algorithms converge uniformly with respect to the granularity of the grid, the polynomial approximation degree p, and the number of levels, provided that the number of smoothing steps, which depends on p, is chosen sufficiently large. Numerical experiments confirm the effectiveness of the proposed schemes and demonstrate that the proposed solvers are convergent in practice, even when some of the theoretical assumptions are not fully satisfied. This is a joint work with Paul Houston (University of Nottingham), Marco Sarti (Politecnico di Milano) and Marco Verani (Politecnico di Milano).

Zhaonan Dong: hp-Version discontinuous Galerkin methods for advection-diffusion-reaction problems on polytopic meshes

In this work, we consider hp-version discontinuous Galerkin finite element method (DGFEM) for the numerical approximation of advection-diffusion-reaction on general computational meshes consisting of polygonal/polyhedral element. In particular, new hp-a priori error bounds are derived in this work which improves the hp-bounds in the work by [P.Houston, C.Schwab, E.Süli - Discontinuous hp-Finite Element Methods for Advection-Diffusion-Reaction Problems, SIAM Journal on Numerical Analysis 39(6):2133-2163, 2002]. The proposed analysis covers the case of equations of changing type and extends to time-dependent parabolic problem for which the hp-a priori error bound is also sharp.

The presented method employs elemental polynomial bases of total degree P defined on the physical space, without the need to map from a given reference or canonical frame. A series of numerical experiments over boundary value problems and time dependent problems highlighting the performance of the proposed method are presented. Moreover, the competitiveness of the DGFEM employing total degree P basis in comparison to the DGFEM employing Q basis on tensor-product elements is studied numerically.

Konstantin Lipnikov: Design principles of the mimetic finite difference schemes

The mimetic finite difference (MFD) method mimics fundamental properties of mathematical and physical systems such as conservation laws, duality of differential operators, and exact mathematical identities of the vector and tensor calculus. The MFD method works on general polygonal and polyhedral meshes. I review design principles underpinning the construction of mimetic finite difference schemes with the focus on consistency and stability conditions. The potential and flexibility of the mimetic discretization framework will be illustrated with challenging examples in Lagrangian gasdynamics, reactive flow and transport in porous media, and nonlinear heat conduction.

Simon Lemaire: A review of Hybrid High-Order methods: formulations, computational aspects, links with other methods

Hybrid High-Order (HHO) methods are formulated in terms of discrete unknowns attached to mesh faces and cells (hence, the term hybrid), and these unknowns are polynomials of arbitrary order k >= 0 (hence, the term high-order). HHO methods are devised from local reconstruction operators and a local stabilization term. The discrete problem is assembled cellwise, and cell-based unknowns can be eliminated locally by static condensation. HHO methods support general meshes, their construction is dimension-independent, they are locally conservative, and they allow for a robust treatment of physical parameters in various situations, e.g., heterogeneous/anisotropic diffusion, quasi-incompressible linear elasticity, and advection-dominated transport. This talk will review HHO methods for a variable-diffusion model problem, detailing both primal and mixed formulations, tackling numerical aspects, and drawing links with other discretization methods from the literature.

Scott Mitchell: VoroCrust: Simultaneous Surface Reconstruction and Volume Meshing with Voronoi cells

I'll describe VoroCrust, the first algorithm for simultaneous surface reconstruction and volumetric Voronoi meshing. By surface reconstruction, I mean that weighted sample points are created on a smooth manifold, and we are tasked with building a mesh (triangulation) containing those points that approximates the surface. By Voronoi meshing, I mean that we create Voronoi cells that are well-shaped polytopal decompositions of the spaces inside and outside the manifold. By "simultaneous", I mean that the surface mesh is the interface of the two volume meshes.

VoroCrust meshes are distinguished from the usual approach of clipping Voronoi cells by the manifold, which results in many extra surface vertices beyond the original samples, and may result in non-planar, non-convex, or even non-star-shaped cells.

The VoroCrust algorithm is similar to the famous "power crust." Unlike the power crust, our output Voronoi cells are unweighted and have good aspect ratio. Moreover, there is complete freedom of how to mesh the volume far from the surface. Most of the reconstructed surface is composed of Delaunay triangles with small circumcircle radius, and all samples are vertices. In the presence of slivers, the reconstruction lies inside the sliver, interpolating between its upper and lower pair of bounding triangles, and introducing Steiner vertices.

Lin Mu: Weak Galerkin Finite Element Methods and Numerical Applications

Weak Galerkin FEMs are new numerical methods that were first introduced by Wang and Ye for solving general second order elliptic PDEs. The differential operators are replaced by their weak discrete derivatives, which endows high flexibility. This new method is a discontinuous finite element algorithm, which is parameter free, symmetric, and absolutely stable. Furthermore, through the Schur-complement technique, an effective implementation of the WG is developed. Several numerical applications will be discussed.

Yann Savoye: "Free at Last": Cage-based Living Geometry

Nowadays, video-based animations of live-actor performances are easier to acquire. Also, 3D Video has reached considerable attentions in visual media productions. In this talk, we focus on the importance of polytopes in the form of volumic cage-based deformers for capturing and reusing surfaces in motion. We define cage-based performance capture as the process of capturing non-rigid surfaces of actors from multi-view using polytopal cages as an underlying deformable model. As a result, registered cage-handles trajectories allow the reconstruction of complex mesh sequences by deforming an enclosed mesh. In the context of vision-based graphics, polytopes offer sufficient degrees of freedom and space abstraction to dynamically capture non-rigid surface motion and to learn space-time shape variability. We illustrate the problem of extracting, acquiring and reusing non-rigid polytopal parametrization in four items: (1) cage-based inverse kinematics, (2) conversion of dynamic surface into cage-based animation, (3) cage-based surface stylization, and (4) cage-based registration. Finally, we demonstrate the benefit of 3D polytopes as reduced domain for performance capture, decoupling motion from geometry.

Jiangguo Liu: C++ Polymorphism for Weak Galerkin Finite Element Methods on Polytopal Meshes

In this talk, we present preliminary results of utilizing C++ polymorphism for implementation of weak Galerkin finite element methods on polytopal meshes. Numerical results obtained from this C++ code for 2-dim Darcy flow computation will be demonstrated. Challenges in 3-dim implementation will be discussed also.

Xiaozhe Hu: Finite Element Multigrid Framework for Mimetic Finite Difference Discretizations

We are interested in the efficient multigrid method of linear systems of equations discretized from the mimetic finite difference (MFD) schemes which work on general unstructured and irregular grids and result in discrete operators that satisfy the exact sequence connecting grad, div and curl operators on the continuous level. We derive such MFD schemes from the standard finite element spaces. Using the finite element framework, we are able to analyze the convergence of the MFD discretizations and construct efficient multigrid methods for the MFD discretizations of elliptic partial differential equations based on the Local Fourier analysis. Finally, we present several numerical tests to demonstrate the robustness and efficiency of the proposed multigrid methods. This is joint work with F. Gaspar, C. Rodrigo, and L. Zikatanov.

N. Sukumar: Numerical integration of homogeneous functions on convex and nonconvex polygons and polyhedra

Accurate integration of polynomial functions over arbitrarily-shaped polygonal and polyhedral domains is required in methods such as the extended finite element method (X-FEM), embedded interface methods, virtual element method, and discontinuous and weak Galerkin methods, just to name a few. The most common approaches to perform this integration have been: tessellation of the domain into simplices; application of Stokes's theorem to reduce the volume integral to a surface integral; and use of moment-fitting methods.

In this talk, I will present a new approach that uses Stokes's theorem and the property of homogeneous functions, whereby it suffices to integrate such functions on the boundary facets of the polytope. For homogeneous polynomials, this approach is used to further reduce the integration to just evaluation of the function and its partial derivatives at the vertices of the polytope. This results in an exact cubature rule for a homogeneous polynomial. Numerical examples in two and three dimensions will be presented to demonstrate the efficacy of the integration scheme, and as an application we consider elastic fracture in 2D using the X-FEM to showcase the capabilities of the method. This is joint-work with Jean Lasserre and Eric Chin.