Finite Element Discontinuous Galerkin Method Research Articles

Hydro-morphodynamic modelling of mangroves imposed by tidal waves using finite element discontinuous Galerkin method

Hydro-morphodynamic modelling of mangroves imposed by tidal waves using finite element discontinuous Galerkin method

Discontinuous Galerkin methods for stochastic Maxwell equations with multiplicative noise

In this paper we propose and analyze finite element discontinuous Galerkin methods for the one- and two-dimensional stochastic Maxwell equations with multiplicative noise. The discrete energy law of the semi-discrete DG methods were studied. Optimal error estimate of the semi-discrete method is obtained for the one-dimensional case, and the two-dimensional case on both rectangular meshes and triangular meshes under certain mesh assumptions. Strong Taylor 2.0 scheme is used as the temporal discretization. Both one- and two-dimensional numerical results are presented to validate the theoretical analysis results.

A High-Order Discontinuous Galerkin Method for the Poro-elasto-acoustic Problem on Polygonal and Polyhedral Grids

The aim of this work is to introduce and analyze a finite element discontinuous Galerkin method on polygonal meshes for the numerical discretization of acoustic waves propagation through poroelastic materials. Wave propagation is modeled by the acoustics equations in the acoustic domain and the low-frequency Biot's equations in the poroelastic one. The coupling is introduced by considering (physically consistent) interface conditions, imposed on the interface between the domains, modeling both open and sealed pores. Existence and uniqueness is proven for the strong formulation based on employing the semigroup theory. For the space discretization we introduce and analyze a high-order discontinuous Galerkin method on polygonal and polyhedral meshes, which is then coupled with Newmark-$\beta$ time integration schemes. A stability analysis both for the continuous problem and the semi-discrete one is presented and error estimates for the energy norm are derived for the semidiscrete problem. A wide set of numerical results obtained on test cases with manufactured solutions are presented in order to validate the error analysis. Examples of physical interest are also presented to test the capability of the proposed methods in practical cases.

Efficient implementation of space-time adaptive ADER-DG finite element method with LST-DG predictor and a posteriori sub-cell WENO finite-volume limiting for simulation of non-stationary compressible multicomponent reactive flows

The present work is devoted to the study of efficient implementation of spacetime adaptive ADER finite element discontinuous Galerkin method with a posteriori correction technique of solutions on subcells by the finite-volume ADER-WENO limiter scheme for simulation of non-stationary compressible multicomponent reactive flows. The multicomponent and reaction properties of the flow are considered in the form of convection-reaction equations. Therefore an effective scheme of splitting the original nonlinear system of algebraic equations of LST-DG predictor was developed to obtain an efficient iterative solution method. This approach is based on the use of the linearity property of the system of equations for the discrete spacetime solution for the system of convection equations, which makes it possible to pre-compute the inverse matrices, and allows the vectorized calculation of the transfer of individual components. The change in the concentrations of the components in the convection-reaction equations and the energy yield in the gas-dynamic equations due to the reactions are calculated using the global iterations. The iterative scheme for the original essentially nonlinear system of equations is split into a sequence of smaller systems. The approach is well applicable for the problems, especially in the case of a large number of components and reactions in the flow.

Two-dimensional frequency-domain visco-elastic full waveform inversion: Parallel algorithms, optimization and performance

Full waveform inversion (FWI) is an appealing seismic data-fitting procedure for the derivation of high-resolution quantitative models of the subsurface at various scales. Full modelling and inversion of visco-elastic waves from multiple seismic sources allow for the recovering of different physical parameters, although they remain computationally challenging tasks. An efficient massively parallel, frequency-domain FWI algorithm is implemented here on large-scale distributed-memory platforms for imaging two-dimensional visco-elastic media. The resolution of the elastodynamic equations, as the forward problem of the inversion, is performed in the frequency domain on unstructured triangular meshes, using a low-order finite element discontinuous Galerkin method. The linear system resulting from discretization of the forward problem is solved with a parallel direct solver. The inverse problem, which is presented as a non-linear local optimization problem, is solved in parallel with a quasi-Newton method, and this allows for reliable estimation of multiple classes of visco-elastic parameters. Two levels of parallelism are implemented in the algorithm, based on message passing interfaces and multi-threading, for optimal use of computational time and the core-memory resources available on modern distributed-memory multi-core computational platforms. The algorithm allows for imaging of realistic targets at various scales, ranging from near-surface geotechnic applications to crustal-scale exploration.

Investigation of the efficiency of using numerical schemes of a high order of accuracy for solving Navier-Stokes and Reynolds equations on unstructured adapted grids

The finite element discontinuous Galerkin method is implemented for solving the Navier-Stokes and Reynolds equations on unstructured adapted grids. A detailed description of the method is given. In problems concerning laminar flow around a cylinder and turbulent flow about a flat plate, solutions with a high order of accuracy are presented. Examples of the calculation of a viscous transonic flow around an isolated airfoil and the subsonic flow around a three-element configuration are considered. These important application problems are solved using the adapted grid technique.