ABSTRACT. We present a family of new mixed finite element methods for the linear elasticity and then develop a robust overlapping domain decomposition method to solve the linear system. Our mixed discretization, preserving the symmetry and the exact $H(div)$ conformity in the stress approximation, can be efficiently implemented by hybridization, which reduces the indefinite system to a symmetric positive-semidefinite system. The condition number of the reduced system, which is characterized by a non-inherited bilinear form, depends not only on the grid size but also on the material parameters. By constructing uniformly stable interpolation operators between the non-nested spaces, we prove that our overlapping domain decomposition method converges uniformly with respect to both the grid size and Poisson's ratio. Numerical experiments are presented to validate our theoretical results.

ABSTRACT. We present a Ritz-Galerkin discretization on sparse grids using prewavelets, which allows to solve elliptic differential equations with variable coefficients for dimension d = 2, 3 and higher dimensions d > 3. To reduce the complexity of the sparse grid discretization matrix, we apply prewavelets and the semi-orthogonality property. An efficient algorithm for matrix vector multiplication using a Ritz-Galerkin discretization is presented. This algorithm is based on standard 1-dimensional restrictions and prolongations, a simple prewavelet stencil, and the classical operator dependent stencil for multilinear finite elements. Simulation results show a convergence of the discretization according to the approximation properties of the finite element space. The linear equation system is solved by a preconditioned conjugate gradient method. Numerical simulation results are presented for a 3-dimensional problem on a curvilinear bounded domain and for a 6-dimensional problem with variable coefficients. Furthermore, simulation results for homogeneous and inhomogeneous boundary conditions are presented.

ABSTRACT. Magnetohydrodynamics (MHD) models describe a wide range of plasma physics applications, from thermonuclear fusion in tokamak reactors to astrophysical models. These models are characterized by a nonlinear system of partial differential equations in which the flow of the fluid strongly couples to the evolution of electromagnetic fields. In this talk, we consider the one-fluid, viscoresistive MHD model in two dimensions. There have been numerous finite-element formulations applied to this problem, and we will briefly discuss the applications of a mixed-method formulation. Here, we consider inf-sup stable elements for the incompressible Navier-Stokes portion of the formulation, Ned\'{e}l\'{e}c elements for the magnetic field, and a second Lagrange multiplier added to Faraday's law to enforce the divergence-free constraint on the magnetic field.

Regardless of the formulation, the discrete linearized systems that arise in the numerical solution of these equations are generally difficult to solve, and require effective preconditioners to be developed. Thus, the final portion of the talk will involve a discussion of monolithic multigrid preconditioners, using an extension of a well-known relaxation scheme from the fluid dynamics literature, Vanka relaxation, to this formulation. To isolate the relaxation scheme from the rest of the multigrid method, we utilize structured grids, geometric interpolation operators, and Galerkin coarse grid operators. Numerical results are shown for the Hartmann flow problem, a standard test problem in MHD.

ABSTRACT. When faced with higher order discretizations, multigrid practitioners often adopt the practice of coarsening in polynomial degree (p) before coarsening in space (h). This process is by nature a geometric one, which is why it is not that frequently implemented in the algebraic multigrid context. We present a framework which aims to reduce non-matrix information that an application needs to provide in order to do hp-multigrid. We focus on nodal continuous Galerkin elements, and demonstrate p-coarsening on multiple levels between arbitrary polynomial orders, as well as the propagation of information necessary to do block smoothing on coarser levels. We show some computational results using an implementation in Trilinos/MueLu.

ABSTRACT. Hyperbolic problems possess solutions of very low regularity as they can have jump discontinuities. Therefore, it is natural to seek $L^2$-norm approximations of the exact solution. The best $L^2$-norm approximation can be computationally obtained by the standard $LL^*$ method but only for a special choice of finite element spaces in the range of $L^*$. Alternatively, the standard $LL^*$ method is used as an intermediary to obtain approximations on more common finite element spaces. This can be achieved in a few ways constituting $LL^*$-type minimization principles and the $(LL^*)^{-1}$ method -- a negative norm approach related to the $LL^*$ method. Furthermore, $LL^*$ and $(LL^*)^{-1}$ methods can be combined with a standard $L^2$-norm least-squares principle resulting in hybrid procedures, which address the minimization of the corresponding graph norm. Here, the different approaches are compared in terms of errors in the obtained approximate solutions.

ABSTRACT. The thermal radiative transfer (TRT) equation describes a statistical behavior of photons interacting with surrounding media. Absorption and reemission in X-ray regime is one of the dominant interaction mechanisms between radiation and the host media, especially in optically thick regions. The absorption and reemission introduces a strong nonlinearity into the original transport equation. Convergence of absorption-reemission physics via simple fixed-point iterations between radiation transport and material energy equation often becomes prohibitively slow.

To remedy slow convergence, we have recently developed a High-Order, Low-Order (HOLO) algorithm, which accelerates the solution of the TRT equation (e.g., HO equation) with Low-order (LO), continuum equations. This algorithm shifts the stiff nonlinearity into a smaller, coarse-level, LO system. A consistently derived LO system can then be solved efficiently using a multigrid preconditioned Newton-Krylov method. Although the HOLO algorithm has originally developed as a discretely-consistent moment-based acceleration scheme, we have recently realized that the algorithm can also be cast as a two-level nonlinear multigrid method based on Full Approximation Scheme (FAS). Viewing our HOLO algorithm as an FAS enables deriving important algorithmic components in terms of the formal multigrid components, including the inter-grid transfer operator, coarse-grid system, and fine-grid relaxation. In this talk, we discuss the algorithmic detail and recent improvements of our method, and we demonstrate applicability of the algorithm with the radiation-hydrodynamics example problems.

ABSTRACT. Conventional adaptive methods depend on the solution being smooth in much of the domain, such that coarse grids provide adequate resolution everywhere but in some localized areas that require higher resolution. Heterogeneity impedes standard adaptive techniques because it is not possible to have accurate solutions without resolving the heterogeneity. We present an adaptive method that can be viewed as a nonlinear multigrid or domain decomposition scheme in which fine levels resolve the heterogeneity but are rarely visited in most of the domain during the nonlinear solution process. This requires judicious use of overlap similar to the highly parallel segmental refinement methods of Adams et al (2016). We will compare and contrast our method with other nonlinear multilevel domain decomposition methods such as ASPIN.

ABSTRACT. Recently, diffusion-based metrics have been developed for protein-protein interaction networks and used for protein function prediction.  In this talk, we focus on the challenges in computing these diffusion-based metrics, especially for large-scale networks.  By exploring the algebraic properties of the distance metrics, we reformulate the computation of distances into solving a series of graph Laplacians systems.  In particular, we develop algebraic multilevel methods for solving the resulting linear systems efficiently.  Applications to the protein-protein networks will be presented and possible generalizations will be discussed. This is a joint work with Lenore Cowen and Junyuan Lin.

ABSTRACT. In this work we study the geometric multigrid method applied to discontinuous Galerkin discretizations of elliptic PDEs. Numerical experiments are given, with a focus on employing high order polynomial basis functions. These experiments illustrate the significant performance potential that multigrid solvers offer. Moreover, preliminary results regarding parallel implementation and performance of this geometric multigrid method will also be presented.