The arbitrary Lagrangian-Eulerian (ALE) method is widely used in the field of compressible multi-material and multi-phase flow problems. In order to implement the indirect ALE approach for the simulation of compressible flow in the context of high order discontinuous Galerkin (DG) discretizations, we present a high order positivity-preserving DG remapping method based on a moving mesh solver in this paper. This remapping method is based on the ALE-DG method developed by Klingenberg et al. [17, 18] to solve the trivial equation $\frac{∂u}{∂t} = 0$ on a moving mesh, which is the old mesh before remapping at $t = 0$ and is the new mesh after remapping at $t = T.$ An appropriate selection of the final pseudo-time $T$ can always satisfy the relatively mild smoothness requirement (Lipschitz continuity) on the mesh movement velocity, which guarantees the high order accuracy of the remapping procedure. We use a multi-resolution weighted essentially non-oscillatory (WENO) limiter which can keep the essentially non-oscillatory property near strong discontinuities while maintaining high order accuracy in smooth regions. We further employ an effective linear scaling limiter to preserve the positivity of the relevant physical variables without sacrificing conservation and the original high order accuracy. Numerical experiments are provided to illustrate the high order accuracy, essentially non-oscillatory performance and positivity-preserving of our remapping algorithm. In addition, the performance of the ALE simulation based on the DG framework with our remapping algorithm is examined in one- and two-dimensional Euler equations.

This paper is concerned with efficient numerical methods for the advection-diffusion equation in a heterogeneous porous medium containing fractures. A dimensionally reduced fracture model is considered, in which the fracture is represented as an interface between subdomains and is assumed to have larger permeability than the surrounding area. We develop three global-in-time domain decomposition methods coupled with operator splitting for the reduced fracture model, where the advection and the diffusion are treated separately by different numerical schemes and with different time steps. Importantly, smaller time steps can be used in the fracture-interface than in the subdomains. The first two methods are based on the physical transmission conditions, while the third one is based on the optimized Schwarz waveform relaxation approach with Ventcel-Robin transmission conditions. A discrete space-time interface system is formulated for each method and is solved iteratively and globally in time. Numerical results for two-dimensional problems with various Péclet numbers and different types of fracture are presented to illustrate and compare the convergence and accuracy in time of the proposed methods with local time stepping.

We extend the pure source transfer domain decomposition method (PSTDDM) to solve the perfectly matched layer approximation of Helmholtz scattering problems in heterogeneous media. We first propose some new source transfer operators, and then introduce the layer-wise and block-wise PSTDDMs based on these operators. In particular, it is proved that the solution obtained by the layer-wise PSTDDM in $\mathbb{R}^2$ coincides with the exact solution to the heterogeneous Helmholtz problem in the computational domain. Second, we propose the iterative layer-wise and block-wise PSTDDMs, which are designed by simply iterating the PSTDDM alternatively over two staggered decompositions of the computational domain. Finally, extensive numerical tests in two and three dimensions show that, as the preconditioner for the GMRES method, the iterative PSTDDMs are more robust and efficient than PSTDDMs for solving heterogeneous Helmholtz problems.

Most existing cell-centered finite volume schemes need to introduce auxiliary unknowns in order to maintain the second-order accuracy when the mesh is distorted or the problem is discontinuous, so interpolation algorithms of auxiliary unknowns are required. Interpolation algorithms are not only difficult to construct, but also bring extra computation. In this paper, an interpolation-free cell-centered finite volume scheme is proposed for the heterogeneous and anisotropic convection-diffusion problems on arbitrary polyhedral meshes. We propose a new interpolation-free discretization method for diffusion term, and two new second-order upwind algorithms for convection term. Most interestingly, the scheme can be adapted to any mesh topology and can handle any discontinuity strictly. Numerical experiments show that this new scheme is robust, possesses a small stencil, and has approximately second-order accuracy for both diffusion-dominated and convection-dominated problems.

The baroclinic-barotropic mode splitting technique is commonly employed in numerical solutions of the primitive equations for ocean modeling to deal with the multiple time scales of ocean dynamics. In this paper, a second-order implicit-explicit (IMEX) scheme is proposed to advance the baroclinic-barotropic split system. Specifically, the baroclinic mode and the layer thickness of fluid are evolved explicitly via the second-order strong stability preserving Runge-Kutta scheme, while the barotropic mode is advanced implicitly using the linearized Crank-Nicolson scheme. At each time step, the baroclinic velocity is first computed using an intermediate barotropic velocity. The barotropic velocity is then corrected by re-advancing the barotropic mode with an improved barotropic forcing. Finally, the layer thickness is updated by coupling the baroclinic and barotropic velocities together. In addition, numerical inconsistencies on the discretized sea surface height caused by the mode splitting are alleviated via a reconciliation process with carefully calculated flux deficits. Temporal truncation error is also analyzed to validate the second-order accuracy of the scheme. Finally, two benchmark tests from the MPAS-Ocean platform are conducted to numerically demonstrate the performance of the proposed IMEX scheme.

We propose two families of nonconforming elements on cubical meshes: one for the −curl∆curl problem and the other for the Brinkman problem. The element for the −curl∆curl problem is the first nonconforming element on cubical meshes. The element for the Brinkman problem can yield a uniformly stable finite element method with respect to the viscosity coefficient $ν.$ The lowest-order elements for the −curl∆curl and the Brinkman problems have 48 and 30 DOFs on each cube, respectively. The two families of elements are subspaces of $H({\rm curl};Ω)$ and $H({\rm div};Ω),$ and they, as nonconforming approximation to $H({\rm gradcurl};Ω)$ and $[H^1 (Ω)]^3,$ can form a discrete Stokes complex together with the serendipity finite element space and the piecewise polynomial space.

In this work, we develop a stochastic gradient descent method for the computational optimal design of random rough surfaces in thin-film solar cells. We formulate the design problems as random PDE-constrained optimization problems and seek the optimal statistical parameters for the random surfaces. The optimizations at fixed frequency as well as at multiple frequencies and multiple incident angles are investigated. To evaluate the gradient of the objective function, we derive the shape derivatives for the interfaces and apply the adjoint state method to perform the computation. The stochastic gradient descent method evaluates the gradient of the objective function only at a few samples for each iteration, which reduces the computational cost significantly. Various numerical experiments are conducted to illustrate the efficiency of the method and significant increases of the absorptance for the optimal random structures. We also examine the convergence of the stochastic gradient descent algorithm theoretically and prove that the numerical method is convergent under certain assumptions for the random interfaces.

In this paper, based on the velocity-pressure formulation of the Stokes eigenvalue problem in $d$-dimensional case $(d=2,3),$ we propose a multigrid discretization of discontinuous Galerkin method using $\mathbb{P}_k−\mathbb{P}_{k−1}$ element $(k≥1)$ and prove its a priori error estimate. We also give the a posteriori error estimators for approximate eigenpairs, prove their reliability and efficiency for eigenfunctions, and also analyze their reliability for eigenvalues. We implement adaptive calculation, and the numerical results confirm our theoretical predictions and show that our method is efficient and can achieve the optimal convergence order $\mathcal{O}(do f^{ \frac{−2k}{d}} ).$

This paper deals with the numerical approximation of a pressure/displacement formulation of the elastoacoustic vibration problem in the axisymmetric case. We propose and analyze a discretization based on Lagrangian finite elements in the fluid and solid domains. We show that the scheme provides a correct approximation of the spectrum and prove quasi-optimal error estimates. We report numerical results to validate the proposed methodology for elastoacoustic vibrations.