10:00 Francisco-Javier Sayas (U of Delaware):
Transient acoustic waves scattered by penetrable obstacles
In this talk I will present some recent results on the dynamics of linear acoustic waves in the time-domain traveling on free space and being scattered by a finite collection of obstacles with homogeneous material properties. The problem can be recast as a system of time domain boundary integral equations, following the pattern of the Costabel-Stephan system in the frequency domain. We will show well-posedness after Galerkin semidiscretization in space as well as convergence estimates for a fully discrete scheme using convolution quadrature in time. We will also show some numerical experiments corroborating the theoretical results. (This is joint work with Tianyu Qiu.)
10:20 Harbir Antil (George Mason University):
Shape Optimization of Shell Structure Acoustics
This talk will provide a rigorous framework for the numerical solution of shape optimization problems in shell structure acoustics. The structure is modeled with Naghdi shell equations, fully coupled to boundary integral equations on a minimally regular surface, permitting the formulation of three-dimensional radiation and scattering problems on a two-dimensional set of reference coordinates. We will provide well-posedness of this model, and Fréchet differentiability of the state with respect to the surface shape. For a class of shape optimization problems we prove existence of optimal solutions under slightly stronger surface regularity assumptions. Finally, adjoint equations are used to efficiently compute derivatives of the radiated field with respect to large numbers of shape parameters, which allows consideration of a rich space of shapes, and thus, of a broad range of design problems.
10:40 Shelvean Kapita (U of Delaware):
Plane Wave Discontinuous Galerkin Methods
We present a study of two residual a posteriori error indicators for the Plane Wave Discontinuous Galerkin (PWDG) method for the Helmholtz equation. In particular we study the h-version of PWDG in which the number of plane wave directions per element is kept fixed. First we use a slight modification of the appropriate a priori analysis to determine a residual indicator. Numerical tests show that this is reliable but pessimistic in that the ratio between the true error and the indicator increases as the mesh is refined. We therefore introduce a new analysis based on the observation that sufficiently many plane waves can approximate piecewise linear functions as the mesh is refined. This allows for an improvement in the residual indicators. Numerical results demonstrate an improvement in the efficiency of the indicators.
11:00 BREAK
11:30 Enrique Otarola (U of Maryland, College Park):
A PDE approach to space-time fractional parabolic problems
We study solution techniques for evolution equations with fractional diffusion and fractional time derivative in a polyhedral bounded domain. The fractional time derivative, in the sense of Caputo, is discretized by a first order scheme and analyzed in a general Hilbert space setting. We show discrete stability estimates which yield an energy estimate for evolution problems with fractional time derivative. The spatial fractional diffusion is realized as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on a semi-infinite cylinder in one more spatial dimension. We write our evolution problem as a quasi-stationary elliptic problem with a dynamic boundary condition, and we analyze it in the framework of weighted Sobolev spaces. The rapid decay of the solution to this problem suggests a truncation that is suitable for numerical approximation. We propose and analyze a first order semi-implicit fully-discrete scheme to discretize the truncation: first degree tensor product finite elements in space and a first order discretization in time. We prove stability and a near optimal a priori error estimate of the numerical scheme, in both order and regularity. regularity
11:50 Abner Salgado (U of Tennessee):
A PDE approach to fractional diffusion: a posteriori error estimation and adaptivity
We derive a computable a posteriori error estimator for the α-harmonic extension problem, which localizes the fractional powers of elliptic operators supplemented with Dirichlet boundary conditions. The derived estimator relies on the solution of small discrete problems on anisotropic cylindrical stars. It exhibits built-in flux equilibration and is equivalent to the error up to data oscillation. A simple adaptive algorithm is designed and numerical experiments reveal a competitive performance.
12:10 Dong Zhou (Temple University):
Mixed finite element methods for a pressure Poisson equation reformulation of the incompressible Navier-Stokes equations
Popular schemes for the incompressible Navier-Stokes equations (NSE), such as projection methods, are efficient but may introduce numerical boundary layers or have limited temporal accuracy due to their fractional step nature. Pressure Poisson equation (PPE) reformulations represent a class of methods that replace the incompressibility constraint by a Poisson equation for the pressure, with a suitable choice of the boundary condition so that the incompressibility is maintained. PPE reformulations of the NSE have important advantages: the pressure is no longer implicitly coupled to the velocity, thus can be directly recovered by solving a Poisson equation, and no numerical boundary layers are generated. In this talk we focus on finite element approach of the Shirokoff-Rosales PPE reformulation. Interestingly, the "electric" boundary conditions provided for the velocity render classical nodal finite elements non-convergent. We thus present an alternative methodology, mixed finite element methods, and demonstrate that this approach allows for arbitrary order of accuracy both in space and in time.
12:30 LUNCH (will be provided)
  1:30 Ragnar Winther (University of Oslo):             KEYNOTE SPEAKER
Local Bounded Cochain Projections and the Bubble Transform
The study of discretizations of Hodge Laplace problems in finite element exterior calculus unifies the theory of mixed finite element approximations of a number of problems in areas like electromagnetism and fluid flow. The key tool for the stability analysis of these discretizations is the construction of projection operators which commute with the exterior derivative and at the same time are bounded in the proper Sobolev norms. Such projections are referred to as bounded cochain projections.

The canonical projections, constructed directly from the degrees of freedom, will commute with the exterior derivative, but unfortunately, they are not properly bounded. On the other hand, bounded cochain projections have been constructed by combining a smoothing operator and the unbounded canonical projection. However, an undesired property of these smoothed projections is that, in contrast to the canonical projections, they are nonlocal.

Therefore, we have recently proposed an alternative construction of bounded cochain projections, which also is local. This construction can be seen as a variant of the well known Clément operator, and it utilizes a double complex structure defined on the macroelements associated the subsimplexes of the grid. In addition, we will also discuss a new tool for analysis of finite element element methods, referred to as the bubble transform. In contrast to all the projections operators above, this transform will lead to projections with bounds which are independent of the polynomial degree of the finite element spaces. As a consequence, this can potentially simplify the analysis of the so-called p-method.

  2:30 BREAK
  3:00 Wujun Zhang (U of Maryland, College Park):
A finite element method for second order linear elliptic equations in non-divergence form
Fully nonlinear elliptic PDEs, including Monge-Ampere equation and Isaac's equation, arise naturally from differential geometry, optimal mass transport, stochastic games and the other fields of science and engineering. In contrast to an extensive PDE literature, the numerical approximation reduces to a few papers. One major difficulty is the notion of viscosity solution which hinges on the maximum principle, instead of a variational principle. In this talk, we consider linear uniformly elliptic equations in non-divergence form, which may be regarded as linearization of fully nonlinear equations. We discuss the design of convergent numerical methods for such equations. We introduce a novel finite element method which satisfies a discrete maximum principle. This property, together with operator consistency, guarantees convergence to the viscosity solution provided that the coefficient matrix and right-hand side are continuous. We also present a discrete version of the Alexandroff-Bakelman-Pucci estimate, and use it to derive convergence rates under suitable regularity assumptions of the coefficient matrix and viscosity solution.
  3:20 Manuel Sanchez (Brown University):
On the accuracy of finite element approximations to a class of interface problems
A second-order accurate finite element method to solve a class of interface problem in two dimensions is presented. Piecewise linear elements are used and hence the stiffness matrix is independent of the interface. However, adding some corrections terms to the right-hand side allows one to recover second order accuracy. We also present some theoretical tools that allow us to analyze other similar methods. As a result, we show that the natural method (which could be thought of as the finite element version of Peskin's method), although non-optimal near the interface, is optimal for points O (h log(1/h))1/2 away from the interface. Finally, we report some numerical tests which illustrate the results. This is joint work with Johnny Guzman and Marcus Sarkis.
  3:40 Xuan Huang (U of Maryland, Baltimore County):
Challenges and Opportunities in Long-Time Simulations of PDEs on Modern Parallel Computing Platforms
Efficient parallel computing is necessary to enable long-time simulations of complex nonlinear models on high-resolution meshes, such as for calcium induced calcium release in a heart cell modeled by a system of advection-diffusion-reaction equations in three space dimensions. A method of line approach with sophisticated time-stepping and Jacobian-Free Newton-Krylov methods is well suited for parallel computing. With demands to push performance further, we need to use modern parallel hardware with hierarchical memory access, data transmission via InfiniBand, and more, whose optimal use is inherently challenging. The extension of the cluster in the UMBC High Performance Computing Facility, which features multiple memory hierarchies in CPUs and GPUs and 60-core Intel Phi accelerators, offers great research opportunities and potentially significant speedup of the computational kernels in simulation tools such as PDE solver for time-dependent problems. This research is joint work with Matthias K. Gobbert, Jonathan Graf, Samuel Khuvis, Matthew Brewster, Bradford E. Peercy, and Zana Coulibaly.
  4:00 BREAK
  4:30 Gideon Simpson (Drexel University):
Relative Entropy Preconditioning for Markov Chain Monte Carlo
One of the challenges in using Markov Chain Monte Carlo methods to sample from a target distribution is finding a good prior distribution. An ideal prior distribution would both be easy to sample from and have a high acceptance rate in the Metropolis step of the algorithm. This latter property ensures that the Markov chain will rapidly explore the configuration space under the target distribution. In this talk, we present work to use functionalized Gaussian priors which are preconditioned to minimize the distance, with respect to relative entropy, to the target measure. This will then be seen to give much more favorable sampling properties than the naive prior.
  4:50 Edward Phillips (U of Maryland, College Park):
Commutator Based Preconditioners for a Mixed Finite Element MHD Formulation
We consider a mixed finite element formulation for the fully coupled magnetohydrodynamics (MHD) equations (Schotzau 2004). This formulation arises from a coupling of two saddle point systems, one corresponding to the Navier-Stokes equations and the other to Maxwell's equations in mixed form. The hydrodynamic unknowns are discretized with an inf-sup stable nodal element pair and the electromagnetic unknowns are discretized with Nedelec's edge elements and nodal elements. We focus on developing preconditioners for the linear systems arising in nonlinear iterations applied to the finite element formulation. By considering block decompositions of these systems, we embed the strong physical coupling inherent to the MHD equations in the Schur complements of the algebraic system. Modifying commutator based strategies introduced for the discretized Navier-Stokes equations and drawing on preconditioners proposed for Maxwell's equations, we develop robust approximations for these Schur complements that lend themselves to fast, algorithmically scalable block preconditioners. We demonstrate the effectiveness of these preconditioners on a series of two- and three-dimensional test problems.
  5:10 Mona Hajghassem (U of Maryland, Baltimore County):
Near-optimal multigrid preconditioning for distributed optimal control constrained by first order ODEs
We present some recent analytic results regarding multigrid preconditioners for a linear-quadratic distributed optimal control problem constrained by a first-order, linear ordinary differential equation (an initial value problem). This work is motivated by an unexpected half-order drop in approximation we encountered in our quest for designing efficient multigrid preconditioners for space-time distributed optimal control problems constrained by linear parabolic equations. Our attempt to understand the source of this suboptimality has led to the study of the simpler case of the ODE-constrained optimal control problem, where a similar drop in approximation order occurs. The work on the ODE-constrained problem required an interesting analysis that we hope to replicate in analyzing the multigrid methods for parabolic-constrained case.
  5:30 Scott Ladenheim (Temple University):
Indefinite Preconditioning for the Coupled Stokes-Darcy System
We consider the coupled Stokes-Darcy system in a specified domain with two subregions. This is a system of partial differential equations describing two flows. In one subregion of the domain, freely flowing fluid is governed by the Stokes equations. In the other subregion there is flow governed by Darcy’s law. The two flows are then coupled together by conditions on the interface of the two flows. The problem is solved numerically using the finite element method. The Stokes domain is discretized using standard, continuous finite element spaces that satisfy an inf-sup condition. In the Darcy domain we consider both continuous functions and discontinuous polynomials (discontinuous Galerkin methods). In both cases the discretization leads to a system of equations that is large, sparse, non-symmetric (and non-symmetrizable in the DG case) and of saddle point form. We propose solving the system of equations using preconditioned GMRES with an indefinite (or constraint) preconditioner which mimics the structure of the original system matrix. We prove that the convergence of GMRES using this preconditioner is bounded independently of the underlying mesh discretization. We present numerical results showing that the indefinite preconditioner outperforms both standard block diagonal and block triangular preconditioners both with respect to iteration count and CPU times.
  5:50 END