# Poster session

Thursday, June 29, 2017 - 3:40pm - 5:20pm

Lind 400

**The Complex of Nonlinear Elasticity and Compatible-Strain FEMs**

Arzhang Angoshtari (George Washington University)

In this poster, we discuss the Hilbert complex of nonlinear elasticity and its application for obtaining a family of mixed finite element methods for nonlinear elasticity called Compatible-Strain mixed FEMs. The Hilbert complex of nonlinear elasticity contains information about the kinematics and the kinetics of deformations and is isomorphic to a variation of the de Rham complex. This isomorphism implies that any property of the de Rham complex has an analogue for the Hilbert complex of nonlinear elasticity. In particular, one can discretize the nonlinear elasticity complex by using the finite element exterior calculus. Mixed finite element methods for nonlinear elasticity are then obtained by using these discrete complexes.**A Dual Mixed Hybridized Finite Element Method for Three-Dimensional Transmission Problems**

Riccardo Sacco (Politecnico di Milano)

In this poster we consider a linear transmission problem in three dimensions to be solved in a polyhedral domain composed of the union of two subdomains, separated by a two-dimensional interface. In each subdomain we solve an elliptic equation with diffusive, advective and reactive terms, whereas across the interface we enforce the balance of the normal flux and a segregation condition. For the numerical approximation we propose a dual mixed hybridized (DMH) finite element method (FEM) based on the Raviart-Thomas finite element space of lowest order. Preliminary numerical experiments indicate that (i) the theoretical convergence estimates valid in the purely diffusive case are satisfied by the novel method applied to the transmission problem when the solution and its normal flux are continuous at the material interface; and (ii) the scheme is stable and accurate also in the case where the solution and its normal flux suffer jump discontinuities at the material interface.**H(div) Mixed Finite Elements of Minimal Dimension on Quadrilaterals and Cuboidal Hexahedra**

Todd Arbogast (The University of Texas at Austin)

We (joint with Maicon R. Correa and Zhen Tao) present two new families of mixed finite elements on quadrilaterals and cuboidal hexahedra. The new spaces use polynomials defined directly on the element and a small number of supplemental basis functions. The supplements are defined on a reference element and mapped by the Piola transform. The new families are inf-sup stable, and they approximate optimally the velocity, pressure, and divergence of the velocity. The spaces are of minimal dimension subject to the approximation properties and finite element conformity (i.e., they lie in H(div) and are constructed locally). The two families give full and reduced H(div) approximation, like Raviart-Thomas and BDM spaces. The two families are identical except for inclusion of a minimal set of vector and scalar polynomials needed for higher order approximation of the divergence of the velocity and pressure.**Transient Waves in Piezoelectric Media: Analysis, Simulation, and Control**

Thomas Brown (University of Delaware)

We study an initial-boundary value problem involving elastic waves propagating in piezoelectric media, such as can be found in cigarette lighters, humidifiers, speakers, and many other everyday objects. A well-posedness result is presented as well as a numerical simulation using finite elements. Additionally we use the problem in the setting of PDE constrained optimization to show control of the waves using the boundary condition on the electric flux.**Asymptotic Approximation of Linear Kinetic Equations with Spectral Methods**

Zheng Chen (Oak Ridge National Laboratory)

We prove some convergence properties for a semi-discrete, moment-based approximation of a model kinetic equation in one dimension. This approximation is equivalent to a standard spectral method in the velocity variable of the kinetic distribution and, as such, is accompanied by standard algebraic estimates of the form $N^{-q}$, where $N$ is the number of modes and $q$ depends on the regularity of the solution. However, in the multiscale setting, we show that the error estimate can be expressed in terms of the scaling parameter $\epsilon$, which measures the ratio of the mean-free-path to the domain in the system. In particular we show that the error in the spectral approximation is $\mathcal{O}(\epsilon^{N+1})$. More surprisingly, the coefficients of the expansion satisfy some super convergence properties. In particular, the error of the $\ell^{th}$ coefficient of the expansion scales like $\mathcal{O}(\epsilon^{2N})$ when $\ell =0$ and $\mathcal{O}(\epsilon^{2N+2-\ell})$ for all $1\leq \ell \leq N$. This result is significant, because the low-order coefficients correspond to physically relevant quantities of the underlying system.**Bernstein-Bezier basis for H(curl) and H(div) Finite Elements on Cubes**

Guosheng Fu (Brown University)

We present a set of Berstein-Bezier basis for the Nedelec-Raviart-Thomas H(curl) and H(div) finite elements on cubes. These basis functions have optimal assembly complexity. They fully respect the differential operators, and enjoy the so-called local exact sequence property.**CutFEM for a Dynamic Interface Problem**

Kyle Dunn (Worcester Polytechnic Institute)

Solving a time-dependent PDE with a dynamic interface is very computationally expensive, particularly without a predetermined interface velocity. The problem becomes even more time-consuming when one must fit a new mesh to the domain each time the interface moves. When using finite element to solve the problem, this can be avoided by using a cut finite element method (CutFEM) to separate the mesh structure from the domain. Using a time-dependent CutFEM method we solve the immersed boundary method on a structured mesh with optimal convergence. We introduce a semi-implicit method to solve this problem and prove it's stability.**A new combined approach using subcell shock capturing and positivity preserving in a high order discontinuous Galerkin scheme applied to 1D shallow water equations**

Alberto Costa Nogueira Junior (IBM Research Brazil)

The Shallow Water Equations (SWE) is a system of hyperbolic PDEs that can be derived from the Navier-Stokes equations under a basic simplifying assumption: the horizontal length scale is considered much greater than the vertical one implying small vertical velocities. The application of SWE to real problems is quite broad covering different hydrological models, specially, runoff, stream networks, lakes and flooding. In the present study, we developed a new high order DG scheme which combines two desirable features: (1) subcell shock capturing based on localized artificial viscosity mechanism and (2) positivity preserving limiting. Numerical experiments with analytical solutions were run to validate the formulation and assess its numerical accuracy. Results revealed that the new scheme is very robust for explicit numerical integration using standard Runge-Kutta schemes (time steps of at most 1e-04). Besides this significant quality, we could also highlight the following notable features of the proposed scheme: remarkable accuracy for very high-order approximation (polynomial degree up to 10) using quite coarse meshes, sharpness in shock representation with minimal artificial viscosity addition, and stability for dry bed solutions.**High Order Entropy Stable Discontinuous Galerkin Methods**

Mohammad Zakerzadeh (RWTH Aachen)

The well-posedness of the entropy weak solutions for scalar conservation laws is a classical result. However, for multidimensional hyperbolic systems, some theoretical and numerical evidence cast doubt on that entropy solutions constitute the appropriate solution paradigm, and it has been conjectured that the more general EMV solutions ought to be considered the appropriate notion of solution. In the numerical framework and building on previous results, we prove that bounded solutions of a certain class of space-time discontinuous Galerkin (DG) schemes converge to an EMV solution. The novelty in our work is that no streamline-diffusion (SD) terms are used for stabilization. While SD stabilization is often included in the analysis of DG schemes, it is not commonly found in practical implementations. In the case of scalar equations, this result can be strengthened and the reduction to the entropy weak solution is obtained. For viscous conservation laws, we extend our framework to general convection-diffusion systems, with both nonlinear convection and nonlinear diffusion, such that the entropy stability of the scheme is preserved. We use a mixed formulation, and handle the difficulties arising from the

nonlinearity of the viscous flux by an additional Galerkin projection operator. We prove the entropy stability of the method for different treatments of the viscous flux, thus unifying and extending some results already existing in the literature.**Finite Element Methods for the Stochastic Allen-Cahn Equation with Gradient-type Multiplicative Noise**

Yi Zhang (University of Notre Dame)

We study finite element approximations of the stochastic Allen-Cahn equation with gradient-type multiplicative noise that is white in time. The sharp interface limit of this stochastic equation formally approximates a stochastic mean curvature flow. Two fully discrete finite element methods based on different time-stepping strategies for the nonlinear term are proposed. We obtain strong convergence with sharp rates with the crucial help of the Holder continuity in time with respect to various spatial norms for the strong solution, and high moment estimates of the strong solution. This is a joint work with Xiaobing Feng and Yukun Li.**Analysis of a Mixed Discontinuous Galerkin Method for Incompressible Magnetohydrodynamics**

Weifeng Qiu (City University of Hong Kong)

In this paper we propose and analyze a mixed DG method for the stationary Magnetohydrodynamics (MHD) equations with two types of boundary (or constraint) conditions. The numerical scheme is based a recent work proposed by Houston et. al. for the linearized MHD. With two novel discrete Sobolev embedding type estimates for the discontinuous polynomials, we provide a priori error estimates for the method on the nonlinear MHD equations. In the smooth case, we have optimal convergence rate for the velocity, magnetic field and pressure in the energy norm, the Lagrange multiplier only has suboptimal convergence order. With the minimal regularity assumption on the exact solution, the approximation is optimal for all unknowns. To the best of our knowledge, this is the first a priori error estimates of DG methods for nonlinear MHD equations.**Kronecker Product Preconditioners for Very High Order Discontinuous Galerkin Methods**

Per-Olof Persson (University of California, Berkeley)

Although the DG method generalizes to arbitrary orders, there are several challenges preventing the use of very high degree polynomial bases. For explicit methods, the CFL condition requires that the time step satisfy approximately dt

We describe an implicit DG method with a tensor-product structure whose computational cost per DOF scales linearly with the degree p. This method requires a tensor-product basis on quadrilateral or hexahedral meshes. The matrix corresponding to the linear system is not explicitly constructed. Fast matrix-vector products are performed as the kernel of the GMRES solver. Such systems are often preconditioned using the block Jacobi preconditioner. To avoid inverting the diagonal blocks and thus incurring the above-mentioned O(p^(3d)) operations, we make use the Kronecker product SVD to approximate this block by a sum of lower-dimensional tensor products.**Ultra-efficient Reduced Basis Method and Its Integration with Uncertainty Quantification**

Yanlai Chen (University of Massachusetts Dartmouth)

Models of reduced computational complexity is indispensable in scenarios where a large number of numerical solutions to a parametrized problem are desired in a fast/real-time fashion. Thanks to an offline-online procedure and the recognition that the parameter-induced solution manifolds can be well approximated by finite-dimensional spaces, reduced basis method (RBM) and reduced collocation method (RCM) can improve efficiency by several orders of magnitudes. The accuracy of the RBM solution is maintained through a rigorous a posteriori error estimator whose efficient development is critical and involves fast eigensolves.

After giving a brief introduction of the RBM/RCM, this poster will show our recent work on significantly delaying the curse of dimensionality for uncertainty quantification, and new fast algorithms for speeding up the offline portion of the RBM/RCM by around 6-fold.**Discontinuous Galerkin Methods, Sensor Observations And Filtering Algorithms For Shallow Water Equations**

Seshu Tirupathi (IBM Research Division)

Flood modeling has generally been considered a boundary value problem with boundary conditions typically specified as hydrographs. More recently, the ability to update the model with observational data from sensors within the domain has gained interest. Observational sources include sensors placed in the river, satellite images GPS equipped drifters. This presentation concentrates on developing algorithms for incorporating sensor observations into the governing shallow water equations for flood modeling.**Divided Difference Estimates for DG Approximations to Nonlinear Scalar Conservation Laws**

Jennifer Ryan (University of East Anglia)

The ability to extract superconvergence relies on higher order information contained in the numerical approximation. Historically, for DG approximations to linear hyperbolic equations, a Smoothness-Increasing Accuracy-Conserving (SIAC) post-processing can improve the order of accuracy from k+1 to 2k+1, when piecewise polynomials of degree k are used in the approximation. This relies on the divided difference of the errors for the DG solution. Obtaining optimal estimates for non-linear hyperbolic conservation laws is not trivial and requires an appropriate choice of basis. In this poster, we present a brief outline of the procedure to obtain optimal error estimates and the difficulties in extending these estimates to multiple-dimensions.**One-dimensional model of blood flow discretized using Runge-Kutta discontinuous Galerkin methods**

Beatrice Riviere (Rice University)**An Adaptive Finite Element Dynamic Phase-Field Fracture Model**

Mallikarjunaiah Muddamallappa (Worcester Polytechnic Institute)

In this work, we describe an efficient finite element treatment of a variational, time-discrete model for dynamic brittle fracture. We propose an efficient numerical scheme based on the bilinear finite elements. For the temporal discretization of the equations of motion, we use generalized \alpha-time integration algorithm, which is implicit and unconditionally stable. To accommodate the crack irreversibility, we use a primal-dual active set strategy, which can be identified as a semi-smooth Newton’s method. It is well known that to resolve the crack-path accurately, the mesh near the crack needs to be very fine, so it is common to use adaptive meshes. We propose a simple, robust, local mesh-refinement criterion to reduce the computational cost. We show that the phase-field based variational approach and adaptive finite-elements provide an efficient procedure for simulating the complex crack propagation including crack-branching. This is joint work with Drs. Christopher Larsen and Marcus Sarkis.