MiniSymposium-3: Advanced Discretizations, Fast Solvers, and Applications
With the development of the computer technologies, detailed simulation of models governed by partial differential equations (PDEs) becomes possible. Advanced discretizations with nice properties, preferred by applications, have been developed. Fast solvers for the resulting large-scale linear and nonlinear systems are important for the efficient algorithms. The scope of this mini-symposium is to present recent advances in discretization methods, fast solvers, and possible applications.
Organizers:
Marcus Sarkis, Worcester Polytechnic Institute
Xuemin Tu, University of Kansas
Saturday, October 2, 2021 at 10:20 – 11:40 am (CST) (Session 1)
Yi Yu
Worcester Polytechnic Institute
10:20 am
A Three-level Extension of Non-overlapping Spectral Additive Schwarz Methods (NOSAS) and its Economic Version
Non-overlapping spectral additive Schwarz methods (NOSAS) were introduced as domain decomposition preconditioners for solving 2D and 3D elliptic problems with highly heterogeneous coefficients inside each subdomain. NOSAS are two-level additive Schwarz methods with coarse space constructed by the local eigenfunctions generated by each local generalized eigenvalue problem. Local problem and local generalized eigenvalue problem use only non-overlapping subdomains and the subdomain iteration is via the coarse space. NOSAS methods have good parallelization properties. The condition number of NOSAS is independent of coefficients and only associated with the local eigenfunctions quantified by a threshold. Also, the global interaction of the coarse problem is associated with the total number of all local eigenfunctions. In this talk, we design and analyze a three-level extension of NOSAS, aiming to reduce the complexity of the global problem. Also, we consider an economic version of local generalized eigenvalue problem, aiming to reduce the complexity of each local generalized eigenvalue problem.
Jinjin Zhang
University of Kansas
10:40 am
BDDC Algorithms for Advection-diffusion Problems with HDG Discretizations
In this talk, a BDDC preconditioned GMRES method is proposed and analyzed for solving the linear system from advection-diffusion equations with HDG discretizations. For large viscosity, the number of iterations is independent of the number of subdomains and depends only slightly on the subdomain problem size when the subdomain size is small enough. However, the convergence deteriorates when the viscosity decreases. These results are similar to those with the standard finite element discretizations. Two examples in two dimensions will be discussed.
Carlos Borges
University of Central Florida
11:00 am
Fast Direct Solver for the Lippmann-Schwinger Equation
A Fast Direct Solver that uses the HODLR fast solver of Sivaram and Darve is applied to solve the Lippmann-Schwinger integral equation for the solution of the Helmholtz problem. The domain is discretized by using an adaptive level-restricted tree structure based on its contrast function. To speed up the solver, we compute the quadratures by using the series representation of the Hankel functions to pre-compute tables that can be used with the tree structure to obtain the far field and local iterations between the discretization points.
Samuel Van Fleet
Iowa State University
11:20 am
A Lax-Wendroff Discontinuous Galerkin Method for Linear Aystems of Hyperbolic Conservation Laws
In this work we develop a Lax-Wendroff discontinuous Galerkin (LxW-DG) method for solving linear systems of hyperbolic partial differential equations (PDEs). The proposed method is a variant of the standard LxW-DG method from the literature. We develop this newly proposed approach in both one and two spatial dimensions and compare the regions of stability to the standard LxW-DG method. We show that compared to the standard LxW-DG method, the modified method has a larger region of stability and has improved accuracy. We demonstrate the properties of this new method by applying it to several numerical test cases.
Saturday, October 2, 2021 at 4:40 – 6:00 pm (CST) (Session 2)
Jue Yan
Iowa State University
4:40 pm
A New Direct Discontinuous Galerkin Method with Interface Correction for Two-dimensional Compressible Navier-Stokes Equations
We propose a new formula for the nonlinear viscous numerical flux and extend the direct discontinuous Galerkin method with interface correction (DDGIC) to compressible Navier-Stokes equations. The new DDGIC framework is based on the observation that the nonlinear diffusion can be represented as a sum of multiple individual diffusion processes corresponding to each conserved variable. A set of direction vectors corresponding to each individual diffusion process is defined and approximated by the average value of the numerical solution at the cell interfaces. The new framework only requires the computation of conserved variables' gradient, which is linear and approximated by the original direct DG numerical flux formula. The proposed method greatly simplifies the implementation, and thus, can be easily extended to general equations and turbulence models. Numerical experiments with $P_1$, $P_2$, $P_3$ and $P_4$ polynomial approximations are performed to verify the optimal $(k+1)^{th}$ high-order accuracy of the method. The new DDGIC method is shown to be able to accurately calculate physical quantities such as lift, drag, and friction coefficients as well as separation angle and Strouhal number. This is joint work with Mustafa Danis.
Guosheng Fu
University of Notre Dame
5:00 pm
Divergence-free (Hybrid) Discontinuous Galerkin Methods for Incompressible Flow Problems
We present a divergence-free (hybrid) discontinuous Galerkin scheme for incompressible flow problems, including incompressible Euler and Naver-Stokes equations, incompressible MHD, and the phase-field model of incompressible two phase flow. Main features of the scheme includes globally divergence-free velocity approximation/exact mass conservation, inherent (minimal amount) numerical dissipation (via DG upwinding) for convection terms which makes the scheme stable in the convection-dominated regime without using extra residual-based stabilizations, efficient linear system solvers via hybridization.)
Thi Thao Phuong Hoang
Auburn University
5:20 pm
Fully Implicit Local Time-stepping Methods for Advection-diffusion Problems in Mixed Formulations
This talk is concerned with numerical schemes for transport problems in heterogeneous porous media, in which different time steps can be used in different parts of the domain. A semi-discrete continuous-in-time formulation of the linear advection-diffusion equation is obtained by using a mixed hybrid finite element method, in which the flux variable represents both the advective and diffusive flux, and the Lagrange multiplier arising from the hybridization is used for the discretization of the advective term. Based on global-in-time and nonoverlapping domain decomposition, we propose two implicit local time-stepping methods to solve the semi-discrete problem. The first method uses the time-dependent Steklov-Poincaré type operator, and the second uses the optimized Schwarz waveform relaxation (OSWR) with Robin transmission conditions. For each method, we formulate a space-time interface problem which is solved iteratively. Each iteration involves solving the subdomain problems independently and globally in time; thus, different time steps can be used in the subdomains. Numerical results for problems with various Peclét numbers and discontinuous coefficients are presented to illustrate the performance of the proposed local time-stepping methods.
Marcus Sarkis
Worcester Polytechnic Institute
5:40 pm
Robust Model Reductions for Elliptic Problems with Heterogeneous High-Contrast Coefficients
The goal of this talk is to present finite element discretizations for second-order elliptic problems with heterogeneous possibly high-contrast coefficients. Based on a class of adaptive domain decomposition preconditioners named Balancing Domain Decomposition with Constraints-BDDC, Variational Multiscale Methods-VMS and Localized Orthogonal Decomposition Methods-LOD, we design robust discretizations and establish a target optimal a priori error energy estimates using only H1 regularity on the solution. This is a joint work with Alexandre Madureira from LNCC Brazil.