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.

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:**

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.

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.

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.

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.

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

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.

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.

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.