MS10: Recent advances in numerical PDE for multi-physics problems


Session A: Oct.1, 10:40am-12:00pm, Classroom Building 121
Session B: Oct.1, 3:10pm-4:30pm, Classroom Building 121
Session C: Oct.1, 5:00pm-6:20pm, Classroom Building 121

10:40am-11:00am (Oct 1)
CLB 121

Juntao Huang,
Texas Tech University

Adaptive multiresolution sparse grid DG: algorithms and its open source C++ package


In this talk, we present our work on adaptive multiresolution sparse grid DG method and its open source C++ package. This method is constructed based on multiwavelets of various kinds, and are demonstrated to be effective in adaptive calculations, particularly for high dimensional applications. Numerical results for Hamilton-Jacobi equations, nonlinear Schrodinger equations and wave equations will be discussed. We will also illustrate the main structure and feature of the open source C++ package.

11:00am-11:20am (Oct 1)
CLB 121
Sanwar Uddin Ahmad,
Virginia State University

Image reconstruction using an adaptive and accelerated iterative gradient type method for Electrical Impedance Tomography problems.
  Electrical impedance tomography (EIT) is an imaging modality that determines the internal conductivity and permittivity distribution based on the voltage measurements made on an object’s surface when currents are applied. Due to its non-invasiveness, non-ionizing characteristics and cost effectiveness, EIT is gaining a lot of attention in recent years. Gradient type methods have been extensively studied and used for solving the EIT problem. However, these methods suffer greatly due to high computational cost at every iteration. In this presentation, we discuss the implementation of an adaptive iterative gradient type method that accelerates the convergence thus reducing the computing cost.

11:20am-11:40am (Oct 1)
CLB 121
Xiaoming He,
Missouri University of Science and Technology

Unconditionally stable numerical methods for Cahn-Hilliard-Navier-Stokes-Darcy system with different densities and viscosities
  In this presentation, we consider the numerical modeling and simulation via the phase field approach for coupled two-phase free flow and two-phase porous media flow of different densities and viscosities. The model consists of the Cahn-Hilliard-Navier-Stokes equations in the free flow region and the Cahn-Hilliard-Darcy equations in porous media that are coupled by several domain interface conditions. It is showed that the coupled model satisfies an energy law. Then we first propose a coupled unconditionally stable finite element method for solving this model and analyze the energy stability for this method. Furthermore, based on the ideas of pressure stabilization and artificial compressibility, we propose an unconditionally stable time stepping method that decouples the computation of the phase field variable, the velocity and pressure of free flow, the velocity and pressure of porous media, hence significantly reduces the computational cost. The energy stability of this decoupled scheme with the finite element spatial discretization is rigorously established. We verify numerically that our schemes are convergent and energy-law preserving. Numerical experiments are also performed to illustrate the features of two-phase flows in the coupled free flow and porous media setting.

11:40am-12:00pm (Oct 1)
CLB 121
Jiangguo (James) Liu
Colorado State University

Fast Numerical Solvers for Subdiffusion Problems with Spatial Interfaces
  Subdiffusion may happen in compartments, e.g., different areas in cell cytoplasm. This leads to subdiffusion problems with spatial interfaces. For temporal discretization, we employ L1 approximation and then a fast evaluation algorithm for Caputo derivative. For spatial discretization, we consider finite volume or similar methods. Based on these, we develop fast numerical solvers for subdiffusion problems with spatial interfaces. Numerical experiments along with brief analysis will be presented to demonstrate the accuracy and efficiency of these new solvers. This is a joint work with Yonghai Li and Boyang Yu at Jilin University (China).

3:10pm-3:30pm (Oct 1)
CLB 121
Carlos N. Rautenberg,
George Mason University

A Spatially Variant Fractional Laplacian Model: Theory and Applications
  We establish a variational definition of the spatially variant fractional Laplacian and determine existence and uniqueness of solutions to the associated elliptic equation. The analysis hinges on the use of non-standard Sobolev spaces with non-Muckenhoupt weights. We further prove increased regularity results for the solution to the elliptic problem and present the use of the fractional operator as a regularizer in imaging applications. The latter leads to the optimal selection of the fractional order in image reconstruction. We finalize the talk with test examples and future research directions.

MS10-B-2 (canceled)
3:30pm-3:50pm (Oct 1)
CLB 121
Xiu Ye,
University of Arkansas at Little Rock

A new CDG method for the Stokes equations with order two superconvergence
  A new conforming discontinuous Galerkin (CDG) finite element method is introduced for solving the Stokes equations. The CDG method gets its name by combining good features of both conforming finite element method and discontinuous finite element method. It has the flexibility of using discontinuous approximation and simplicity in formulation of the conforming finite element method. In this method, discontinuous Pk element is used for velocity and continuous Pk+1 element is used for pressure. This new CDG method is not only stabilizer-free but also has the convergence rate two order higher than the optimal order for velocity. Numerical tests are provided. This is a joint work with Shangyou Zhang at University of Delaware.

3:50pm-4:10pm (Oct 1)
CLB 121
Lin Mu,
University of Georgia

High-order IPDG Method for Anisotropic Diffusion Equations
  In this talk, we present an interior penalty discontinuous Galerkin finite element scheme for solving diffusion problems with strong anisotropy arising in magnetized plasmas for fusion applications. In such application, the anisotropy is introduced by strong magnetic fields. The heat conduction along the parallel field direction may at the order 106 (boundary region) to 1012 (core region) larger than that along perpendicular direction. Due to the high anisotropy ratio, the errors in the parallel may significantly affect the error in the perpendicular direction and thus cause numerical pollution. One possible way is to perform the simulation on the aligned mesh. However, for our interested far scrape off layer region, the filed aligned mesh is almost impossible to use. In order to handle the high geometry fidelity and relax the burden in mesh generation, we propose the high order discontinuous Galerkin methods on the non-aligned mesh together with the efficient linear solver of auxiliary space preconditioner. We demonstrate the accuracy produced by the high-order scheme and efficiency in the preconditioning technique, which is robust to the mesh size and anisotropy of the problem. Several numerical tests are provided to validate the proposed algorithm.

4:10pm-4:30pm (Oct 1)
CLB 121
Seulip Lee,
University of Georgia

A stable enriched Galerkin method for Brinkman problem
  In this work, we present a stable enriched Galerkin (EG) method for the Brinkman equations with small viscosity. The discrete inf-sup condition for numerical velocity and pressure has been successfully achieved by recent work on new EG methods for the Stokes equations whose numerical velocity is a discontinuous piecewise linear function. We extend the EG methods to the Brinkman equations and apply a velocity reconstruction operator in the discretization to obtain uniform energy errors as viscosity approaches zero. Numerical analysis shows that our EG method guarantees the optimal convergence rates, and numerical experiments validate the theoretical results.

MS10-C-1 (canceled)
5:00pm-5:20pm (Oct 1)
CLB 121
Maxim Olshanskii,
University of Houston

A finite element modeling of two-phase variable density surface fluids
  This talk reviews a continuum-based model for the process of phase separation in multicomponent lipid membranes exhibiting lateral fluidity. We further introduce a finite element method for solving surface fluid and surface phase-field equations. The models and methods are combined to deliver a finite element method for a thermodynamically consistent phase-field model for surface two-phase fluid with variable density and viscosity. A stable linear splitting approach is introduced and available numerical analysis results are presented. We finally discuss successes and failures of the model to reproduce in vitro experiments with multicomponent vesicles of different lipid compositions.

5:20pm-5:40pm (Oct 1)
CLB 121
Slimane Adjerid,
Virginia Tech

A Stable Immersed Discontinuous Galerkin Method for Wave Propagation in Heterogeneous Acoustic Elastic Media
  Immersed finite element methods are applied to solve interface problems on interface-independent meshes that allow interface elements that are cut by the interface. Here we propose an immersed discontinuous Galerkin (DG) method to solve acoustic-elastic interface problems on Cartesian meshes with interface elements that consist of a combination of fluids and solids separated by interfaces. These problems are modeled by different PDE systems that are coupled by jump conditions across the interfaces. We present a stable weak DG formulation combined with a piecewise polynomial immersed finite element (IFE) space. The IFE space is such that on each interface element we use a piecewise polynomial space satisfying the interface jump conditions while on non-interface elements we use standard polynomial spaces. We discuss the stability of the method and a time-marching algorithm. We conclude with several numerical examples showing the performance of our method by solving problems of wave propagation in heterogeneous elastic acoustic media.

5:40pm-6:00pm (Oct 1)
CLB 121
Weihua Geng,
Southern Methodist University

A Cartesian FMM-accelerated Galerkin boundary integral Poisson-Boltzmann solver
  The Poisson-Boltzmann model is an effective and popular approach for modeling solvated biomolecules in continuum solvent with dissolved electrolytes. In this paper, we report our recent work in developing a Galerkin boundary integral method for solving the linear Poisson-Boltzmann (PB) equation. The solver has combined advantages in accuracy, efficiency, and memory usageas it applies a well-posed boundary integral formulation to circumvent many numerical difficulties associated with the PB equation and uses an O(N) Cartesian Fast Multipole Method (FMM) to accelerate the GMRES iteration. In addition, special numerical treatments such as adaptive FMM order, block diagonal preconditioners, Galerkin discretization, and Duffy's transformation are combined to improve the performance of the solver, which is validated on benchmark Kirkwood's sphere and a series of testing proteins.

6:00pm-6:20pm (Oct 1)
CLB 121
Johannes Tausch,
Southern Methodist University

A fast method for evaluating volume potentials in the Galerkin boundary element method 
  We discuss a new algorithm for volume potentials that arise in boundary element methods for elliptic PDEs. The approach is to apply a modified fast multipole method for a boundary concentrated volume mesh. If h is the meshwidth of the boundary, then the volume is discretized using nearly O(h-2) degrees of freedom, and the algorithm computes potentials in nearly O(h-2) complexity. Here nearly means that logarithmic terms of h may appear. Thus the complexity of volume potentials calculations is of the same asymptotic order as boundary potentials. For sources and potentials with sufficient regularity the parameters of the algorithm can be designed such that the error of the approximated potential converges at any specified rate O(hp). The accuracy and effectiveness of the proposed algorithms are demonstrated for potentials of the Poisson equation in three dimensions.