Purdue University, Department of Mathematics, July 7-8, 2017

**9 am – 9:15 am: ** Opening Remarks

**9:15 am – 9:55 am: **Mary Wheeler, Adaptive Enriched Galerkin Methods for Miscible Displacement in Porous Media.

**10 am – 10:40 am: **Xiaobing Feng, Continuous and discontinuous Galerkin methods for non-divergence form linear elliptic PDEs with applications to Bellman equations

**10:45 am - 11:15 am:** Break

**11:15 am - 11:55 am: **Doug Arnold, The Regge family of finite elements

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**12 pm - 2 pm: **Poster session (lunch provided)

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**2 pm – 2:40 pm: **Ricardo Nochetto, Two-Scale FEMs for Non-Variational Elliptic PDEs and Pointwise Convergence Rates.

**2:45 pm – 3:25 pm:** Junping Wang, Primal-Dual Weak Galerkin Finite Element Methods for PDEs

**3:30 pm -4 pm: **Break

**4 pm - 4:40pm:** Sue Brenner, $C^0$ Interior Penalty Methods

**9 am - 9:40 am:** Bernardo Cockburn, Superconvergence by M-decompositions.

**9:45 am – 10:25 am:** Zhiqiang Cai, A Posteriori Error Estimators of Residual and Recovery Types.

**10:30 am - 11 am:** Break

**11 am - 11:40 am: **Ragnar Winther, The bubble transform and its applications.

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**11:45 am – 1:30 pm: **Lunch

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**1:30 pm - 2:10 pm:** Paola Pietra, A Plane Wave Virtual Element Method for the Helmholtz Problem.

**2:15 pm – 2:55 pm:** Dongwoo Sheen, Hybrid two-step finite element method: flux approximation and recovery of primary variable.

**3 pm - 3:30 pm:** Break

**3:30 pm – 4:10 pm: **Todd Arbogast, Mixed Methods for Two-Phase Darcy-Stokes Mixtures of Partially Melted Materials with Regions of Zero Porosity

**6:30 pm - 9:30 pm: **Banquet, Hilton Garden Inn, 356 E. State St, West Lafayette

**Title:** Mixed Methods for Two-Phase Darcy-Stokes Mixtures of Partially Melted Materials with Regions of Zero Porosity

**Speaker:** Todd Arbogast (Joint with Marc A. Hesse and Abraham L. Taicher)

**Abstract: **The Earth's mantle, or an ice sheet, involves a deformable solid matrix phase within which a second phase, a fluid, may form due to melting processes. The mechanics of this system is modeled as a dual-continuum, with at each point of space the solid matrix being governed by a Stokes system and the fluid melt, if it exists, being governed by a Darcy law. This system is mathematically degenerate when the porosity (volume fraction of fluid) vanishes. We develop the variational framework needed for accurate approximation of this Darcy-Stokes system, even when there are regions of positive measure where only one phase exists. We then develop an accurate mixed finite element method for solving the system and show some numerical results.

**Title:** The Regge family of finite elements

**Speaker:**Doug Arnold, University of Minnesota.

**Abstract:** Over the past decade there has been a great deal of interest in compatible or structure-preserving discretizations of PDE, that is, discretizations which exactly retain certain key geometric or algebraic properties of the continuous problem at the discrete level. Structure-preserving finite element methods have been developed for differential forms, such as arise in electromagnetic and flow applications, and for stress fields in solid mechanics. In this talk we will introduce a new family of finite element spaces, devised for discretization of another sort of field important in applications: Riemannian metrics and other symmetric covariant tensors of rank 2. In the lowest order case these new finite elements are intimately related to discrete metrics introduced by Tullio Regge in 1961 for the study of general relativity; hence their name. Special cases of the new elements have connections to classical plate bending elements and to a recent novel approach to elasticity as well. This work is joint with Lizao Li and will appear in his thesis.

**Title:** $C^0$ Interior Penalty Methods

**Speaker:** Sue Brenner, LSU

**Abstract:** $C^0$ interior penalty methods are discontinuous Galerkin methods for fourth order problems that are based on standard Lagrange finite element spaces for second order problems. In this talk we will discuss the a priori and a posteriori error analyses of these methods for fourth order elliptic boundary value problems and elliptic variational inequalities.

**Title:** A Posteriori Error Estimators of Residual and Recovery Types

**Speaker:** Zhiqiang Cai, Purdue University

**Abstract:** Adaptive mesh refinement (AMR) algorithms are one of two necessary tools for grand challenging problems in scientific computing. Reliability of com- puter simulations is responsible for accurate computer predictions/designs. Efficient and reliable a posteriori error estimation is the key for success of AMR algorithms and the reliability of computer predictions/designs.

During the past four decades, the a posteriori error estimation has been extensively studied, and impressive progress has been made. However, due to its extreme difficulty, this important research field of computational science and engineering remains wide open. In this talk, I will describe our recent work on the residual- and the recovery-based error estimators.

**Title: **Superconvergence by M-decompositions

**Speaker:** Bernardo Cockburn, University of Minnesota

**Abstract: **We use the theory of M-decompositions to systematically construct superconvergent HDG methods and their corresponding mixed methods for polygonal or polyhedral element of general shape.

**Title:** Continuous and discontinuous Galerkin methods for non-divergence form linear elliptic PDEs with applications to Bellman equations

**Speaker:** Xiaobing Feng, University of Tennessee

**Abstract:** In this talk I shall present some newly developed continuous Galerkin (CG or finite element) methods and discontinuous Galerkin (DG) methods for approximating strong solutions of a class of linear elliptic PDEs in non-divergence form whose leading coefficients are only continuous or even only belong to the space of vanishing mean oscillation (VMO). Such PDEs are building blocks of fully nonlinear (Hamilton-Jacobi-) Bellman equations arising from stochastic optimal control and financial mathematics. The proposed numerical methods can use either C^0 or L^2 finite element spaces, they are simple to implement and can be done using standard finite element or DG codes. On the other hand, the convergence analysis of the methods is quite involved and very technical, it requires to establish finite element and DG discrete Calderon-Zygmund estimates, which will be discussed in detail in the talk. Numerical experiments will be presented to demonstrate the effectiveness of the proposed CG and DG methods. This talk is based on some recent joint works with Michael Neilan of University of Pittsburgh and Stefan Schnake of the University of Tennessee.

**Title:** Two-Scale FEMs for Non-Variational Elliptic PDEs and Pointwise Convergence Rates

**Speaker:** Ricardo Nochetto, University of Maryland (Joint work with Wenbo Li, Dimitris Ntogkas and Wujun Zhang.)

**Abstract:** We show that the finite element method (FEM) is able to approximate non-variational elliptic PDEs provided we add a larger scale ε to the usual meshsize h. We use the ε-scale to compute centered second differences of continuous functions which are piecewise linear at the h-scale, thereby replacing the notion of wide stencil while enforcing monotonicity without stringent mesh restrictions. We apply this basic principle to linear PDEs in non-divergence form, the Monge-Ampere equation, and a fully nonlinear PDE for the convex envelope. The scale ε plays a role similar to a finite horizon for integro-differential operators.

We derive pointwise error estimates for all three cases exploiting the separation of scales. A fundamental tool is a novel discrete Alexandroff estimate for continuous piecewise linear functions which states that the max-norm of their negative part is controlled by the Lebesgue measure of the sub-differential of their convex envelope at the contact nodes. We also develop a discrete Alexandroff-Bakelman-Pucci estimate which controls the Lebesgue measure of the sub-differential in terms of the discrete Laplacian via gradient jumps.

**Title: **A Plane Wave Virtual Element Method for the Helmholtz Problem

**Speaker:** Paola Pietra, Istituto di Matematica Applicata e Tecnologie Informatiche, CNR, Italy (Joint work Ilaria Perugia, University of Vienna and Alessandro Russo, University of Milano Bicocca.

**Abstract:** Concerned with the time-harmonic wave propagation governed by the Helmholtz equation, we present a novel Galerkin approximation that can deal with general polygonal partitions. Virtual element methods have been recently introduced as extension of finite elements to general polygonal decompositions for different classes of definite and semidefinite problems. Here we design and analyse a method for an indefinite problem. Because of the oscillatory behavior of solutions to the Helmholtz equation, methods that incorporate information about the solution in the form of plane waves have received attention in the last years. The new method presented here is based on inserting plane wave basis functions within the VEM framework aiming at constructing an H1-conforming, high-order method for the discretisation of the Helmholtz problem.

The main ingredients of this plane wave-VEM (PW-VEM) are: *i:* a low frequency space made of VEM functions, whose basis functions are not explicitly computed in the element interiors; *ii:* a proper local projection operator onto the high-frequency space, made of plane waves; *iii:* a proper stabilization term. Convergence of the $h$-version of the PW-VEM is given and numerical results testing its performance on general polygonal meshes are presented.

**Title: **Hybrid two-step finite element method: flux approximation and recovery of primary variable

**Speaker:** Dongwoo Sheen, Department of Mathematics, Seoul National University

**Abstract:** We present a new two--step method based on the hybridization of mesh sizes in the traditional mixed finite element method. On a coarse mesh, the primary variable is approximated by a standard Galerkin method, whose computational cost is very low. Then, on a fine mesh, an $H({\rm div})$ projection of the dual variable is sought as an accurate approximation for the flux variable.

Our method does not rely on the framework of traditional mixed formulations, the choice of pair of finite element spaces is, therefore, free from the requirement of inf-sup stability condition. More precisely, our method is formulated in a fully decoupled manner, still achieving an optimal error convergence order.

This leads to a computational strategy much easier and wider to implement than the mixed finite element method. Additionally, the independently posed solution strategy allows to use different meshes as well as different discretization schemes in the calculation of the primary and flux variables.

We also present a method of improvement on the computed approximate solutions using iterative methods. The decrease in the error can be estimated and it can be used whether another iteration is needed. Also, a simple scheme will be introduced to obtain the approximate solutions for the primary variable from the flux approximation. The approximate solution for the primary variable has the same order of accuracy as the flux approximations. We provide numerical examples that conform the efficiency and convergence of our methods.

**Title: **Primal-Dual Weak Galerkin Finite Element Methods for PDEs

**Speaker: **Junping Wang, National Science Foundation

**Abstract:** This talk will introduce a primal-dual finite element method for variational problems where the trial and test spaces are different. The essential idea behind the primal-dual method is to formulate the original problem as a PDE-constrained minimization problem where the objective function measures the smoothness of the approximating functions. The corresponding Euler-Lagrange formulation involves the primal (original) equation and its dual with homogeneous data. The two equations are linked together by using properly-defined stabilizer/smoother commonly used in the weak Galerkin finite element methods. The primal-dual method will be discussed for three type of model problems: (1) second order elliptic equation in nondivergence form and Fokker-Planck equation, (2) steady-state linear convection equations, and (3) elliptic Cauchy problems.

Weak Galerkin (WG) is a finite element method for PDEs where the differential operators (e.g., gradient, divergence, curl, Laplacian etc.) in the weak forms are approximated by discrete 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 weak forms for the underlying PDEs. Weak Galerkin is a natural extension of the classical Galerkin finite element method with advantages in many aspects. The goal of this talk is to demonstrate some of these advantages for several PDEs that are difficult to handle by using classical finite elements.

**Title: **Adaptive Enriched Galerkin Methods for Miscible Displacement in Porous Media

**Speaker:** Mary F. Wheeler, Sanghyun Lee, Young-Ju Lee

**Abstract:** Miscible displacement of one fluid by another in a porous medium has attracted considerable attention in subsurface modeling with emphasis on enhanced oil recovery applications. Here flow instabilities arising when a fluid with higher mobility displaces another fluid with lower mobility is referred to as viscous fingering. The latter has been the topic of major physical and mathematical studies for over half a century. Recently, viscous fingering has been applied for proppant-filled hydraulic fracture propagation to efficiently transport the proppant to the tip of fractures. The governing mathematical system that represents the displacement of the fluid mixtures consists of pressure, velocity, and concentration.

Here we present a novel approach to the simulation of miscible displacement by employing an adaptive enriched Galerkin finite element methods (EG) coupled with entropy residual stabilization for transport. EG is formulated by enriching the conforming continuous Galerkin finite element method (CG) with piecewise constant functions. EG provides locally and globally conservative fluxes, which is crucial for coupled flow and transport problems. Moreover, EG has fewer degrees of freedom in comparison with discontinuous Galerkin (DG) and an efficient flow solver has been derived which allows for higher order schemes. We have shown theoretically and computationally that a robust preconditioner can be achieved if one adds pre- and post smoothings to a block preconditioner involving CG and jumps in the discontinuous piecewise constants. Dynamic adaptive mesh refinement is applied in treating geological material discontinuities.

An additional advantage of EG is that only those subdomains that require local conservation need to be enriched with a treatment of high order non-matching grids. Our high order EG transport system is coupled with an entropy viscosity residual stabilization method introduced in to avoid spurious oscillations near shocks. Instead of using limiters and non-oscillatory reconstructions, this method employs the local residual of an entropy equation to construct the numerical diﬀusion, which is added as a nonlinear dissipation to the numerical discretization of the system. The amount of numerical diﬀusion added is proportional to the computed entropy residual. This technique is independent of mesh and order of approximation and has been shown to be efficient and stable in solving many physical problems with CG. Finally we note that it is crucial to have dynamic mesh adaptivity in order to reduce computational costs for large-scale three dimensional applications; both for flow and transport. We employ the entropy residual for dynamic adaptive mesh reﬁnement to capture the moving interface between the miscible ﬂuids. It Our computational results indicate that the entropy residual can be used as a efficient posteriori error indicator.

**Title: **The bubble transform and its applications

**Speaker:** Ragnar Winther, University of Oslo

**Abstract:** In 2015 Rick Falk and I introduced the bubble transform, as a new tool for analyzing finite element methods. This transform is defined with respect to a domain and a given mesh, but the construction is independent of any finite element space. The purpose of this linear map, which is bounded both in L2 and H1, is to decompose functions into ”local bubbles,” and a key property is that it is invariant on the the natural C0 finite element spaces associated the mesh. In this talk we will present the construction of the bubble transform, and discuss potential applications to to the construction of local projection operators which are uniformly bounded in the polynomial degree, and the study of condition numbers of local bases and frames. Furthermore, we will discuss a generalization of the transform into commuting decompositions of the associated de Rham complex.

This is joint work with Richard S. Falk.