Discontinuous Galerkin method
View article: Tandem: An Open-Source High-Performance Computing Volumetric Software to Model Sequences of Earthquakes and Aseismic Slip Across Complex Fault Systems
Tandem: An Open-Source High-Performance Computing Volumetric Software to Model Sequences of Earthquakes and Aseismic Slip Across Complex Fault Systems Open
Simulating sequences of earthquakes and aseismic slip (SEAS) on realistic, 3D fault systems remains a computational challenge. Volumetric approaches offer the necessary physical flexibility to handle complex geometries and heterogeneous of…
View article: LISFLOOD-FP 8.2: GPU-accelerated multiwavelet discontinuous Galerkin solver with dynamic resolution adaptivity for rapid, multiscale flood simulation
LISFLOOD-FP 8.2: GPU-accelerated multiwavelet discontinuous Galerkin solver with dynamic resolution adaptivity for rapid, multiscale flood simulation Open
The second-order discontinuous Galerkin (DG2) solver of the two-dimensional shallow water equations in the raster-based LISFLOOD-FP 8.0 hydrodynamic modelling framework is mostly suited for predicting small-scale transients that emerge in …
View article: Study of a New Mixed Weak Galerkin Formulation for the Electric Field
Study of a New Mixed Weak Galerkin Formulation for the Electric Field Open
In this paper, we present a new mixed weak Galerkin FEM for Maxwell’s equations in the primary electrostatic field–Lagrange multiplier. Our numerical scheme is equipped with stable finite elements composed of polynomials of degree ℓ for th…
View article: A reconstructed discontinuous approximation for distributed elliptic control problems
A reconstructed discontinuous approximation for distributed elliptic control problems Open
In this paper, we present and analyze an internal penalty discontinuous Galerkin method for the distributed elliptic optimal control problems. It is based on a reconstructed discontinuous approximation which admits arbitrarily high-order a…
View article: A reconstructed discontinuous approximation for distributed elliptic control problems
A reconstructed discontinuous approximation for distributed elliptic control problems Open
In this paper, we present and analyze an internal penalty discontinuous Galerkin method for the distributed elliptic optimal control problems. It is based on a reconstructed discontinuous approximation which admits arbitrarily high-order a…
View article: Inf-sup stable space–time Local Discontinuous Galerkin method for the heat equation
Inf-sup stable space–time Local Discontinuous Galerkin method for the heat equation Open
We propose and analyze a space–time Local Discontinuous Galerkin method for the approximation of the solution to parabolic problems. The method allows for very general discrete spaces and prismatic space–time meshes. Existence and uniquene…
View article: Can Explicit Subgrid Models Enhance Implicit LES Simulations? A GPU-Oriented High-Order-Solver Perspective
Can Explicit Subgrid Models Enhance Implicit LES Simulations? A GPU-Oriented High-Order-Solver Perspective Open
High-order Discontinuous Galerkin (DG) methods offer excellent accuracy for turbulent flow simulations, especially when implemented on GPU-oriented architectures that favor very high polynomial orders. On modern GPUs, high-order polynomial…
View article: Can Explicit Subgrid Models Enhance Implicit LES Simulations? A GPU-Oriented High-Order-Solver Perspective
Can Explicit Subgrid Models Enhance Implicit LES Simulations? A GPU-Oriented High-Order-Solver Perspective Open
High-order Discontinuous Galerkin (DG) methods offer excellent accuracy for turbulent flow simulations, especially when implemented on GPU-oriented architectures that favor very high polynomial orders. On modern GPUs, high-order polynomial…
View article: Error analysis of a first-order DoD cut cell method for 2D unsteady advection
Error analysis of a first-order DoD cut cell method for 2D unsteady advection Open
View article: A Simple Error Estimate of Discontinuous Galerkin Methods for Elliptic Equations with Low Regularity
A Simple Error Estimate of Discontinuous Galerkin Methods for Elliptic Equations with Low Regularity Open
In this work, we develop a low-regularity error analysis for the interior-penalty discontinuous Galerkin (IPDG) method, incorporating numerical fluxes originally proposed by Brezzi et al. [Numer. Methods Partial Differential Equations, 16 …
View article: A new analytical technique of the fully implicit Crank-Nicolson discontinuous Galerkin method for the Ginzburg-Landau Model
A new analytical technique of the fully implicit Crank-Nicolson discontinuous Galerkin method for the Ginzburg-Landau Model Open
In this paper, a fully implicit Crank-Nicolson discontinuous Galerkin method is proposed for solving the Ginzburg-Landau equation. By leveraging a novel analytical technique, we rigorously establish the unique solvability of the constructe…
View article: Enriched Galerkin Method for Navier-Stokes Equations
Enriched Galerkin Method for Navier-Stokes Equations Open
This paper presents an enriched Galerkin (EG) finite element method for the incompressible Navier--Stokes equations. The method augments continuous piecewise linear velocity spaces with elementwise bubble functions, yielding a locally cons…
View article: Efficient and Scalable Finite-Element Magnetotelluric Modeling on High-Order Meshes
Efficient and Scalable Finite-Element Magnetotelluric Modeling on High-Order Meshes Open
Summary Three-dimensional (3-D) forward modeling of magnetotelluric (MT) data remains a computationally challenging task, particularly when accurate broadband MT responses are simulated for real-world problems that often involve complex mu…
View article: Stationarity preservation and the low Mach number behaviour of the Discontinuous Galerkin method on Cartesian grids
Stationarity preservation and the low Mach number behaviour of the Discontinuous Galerkin method on Cartesian grids Open
Due to added numerical stabilization (diffusion), the stationary states of numerical methods for hyperbolic problems need not be consistent discretizations of those of the PDEs. A closely related phenomenon is the lack of consistency of co…
View article: Stationarity preservation and the low Mach number behaviour of the Discontinuous Galerkin method on Cartesian grids
Stationarity preservation and the low Mach number behaviour of the Discontinuous Galerkin method on Cartesian grids Open
Due to added numerical stabilization (diffusion), the stationary states of numerical methods for hyperbolic problems need not be consistent discretizations of those of the PDEs. A closely related phenomenon is the lack of consistency of co…
View article: Positivity-Preserving Hybridizable Discontinuous Galerkin Scheme for Solving PNP Model
Positivity-Preserving Hybridizable Discontinuous Galerkin Scheme for Solving PNP Model Open
We introduce a hybridizable discontinuous Galerkin (HDG) scheme for solving the Poisson–Nernst–Planck (PNP) equations. The log-density formulation as introduced by Metti et al. in their paper “Energetically stable discretizations for charg…
View article: Enhancing physical consistency in stochastic optimization for adjoint-based inverse problems: Application to compressible RANS simulations in the discontinuous Galerkin framework
Enhancing physical consistency in stochastic optimization for adjoint-based inverse problems: Application to compressible RANS simulations in the discontinuous Galerkin framework Open
View article: High-order Nodal Space-time Flux Reconstruction Methods for Hyperbolic Conservation Laws on Curvilinear Moving Grids
High-order Nodal Space-time Flux Reconstruction Methods for Hyperbolic Conservation Laws on Curvilinear Moving Grids Open
High-order nodal space-time flux reconstruction (STFR) methods have been developed to solve hyperbolic conservation laws on curvilinear moving grids. Unlike the method-of-lines approach for moving domain simulation, the grid velocity is im…
View article: High-order Nodal Space-time Flux Reconstruction Methods for Hyperbolic Conservation Laws on Curvilinear Moving Grids
High-order Nodal Space-time Flux Reconstruction Methods for Hyperbolic Conservation Laws on Curvilinear Moving Grids Open
High-order nodal space-time flux reconstruction (STFR) methods have been developed to solve hyperbolic conservation laws on curvilinear moving grids. Unlike the method-of-lines approach for moving domain simulation, the grid velocity is im…
View article: Isogeometric fluid-structure interaction using a mixed continuous/discontinuous Galerkin scheme
Isogeometric fluid-structure interaction using a mixed continuous/discontinuous Galerkin scheme Open
A mixed continuous / discontinuous Galerkin scheme is introduced for the simulation of fluid-structure interaction problems in an isogeometric analysis framework. The properties of Non-Uniform Rational B-Spline basis functions are leverage…
View article: Improved $L^2$-error estimates for the wave equation discretized using hybrid nonconforming methods on simplicial meshes
Improved $L^2$-error estimates for the wave equation discretized using hybrid nonconforming methods on simplicial meshes Open
We present improved $L^2$-error estimates on the time-integrated primal variable for the wave equation in its first-order formulation. The space discretization relies on a hybrid nonconforming method, such as the hybridizable discontinuous…
View article: Improved $L^2$-error estimates for the wave equation discretized using hybrid nonconforming methods on simplicial meshes
Improved $L^2$-error estimates for the wave equation discretized using hybrid nonconforming methods on simplicial meshes Open
We present improved $L^2$-error estimates on the time-integrated primal variable for the wave equation in its first-order formulation. The space discretization relies on a hybrid nonconforming method, such as the hybridizable discontinuous…
View article: Large-scale Multigrid with Adaptive Galerkin Coarsening
Large-scale Multigrid with Adaptive Galerkin Coarsening Open
We propose a robust, adaptive coarse-grid correction scheme for matrix-free geometric multigrid targeting PDEs with strongly varying coefficients. The method combines uniform geometric coarsening of the underlying grid with heterogeneous c…
View article: Isogeometric fluid-structure interaction using a mixed continuous/discontinuous Galerkin scheme
Isogeometric fluid-structure interaction using a mixed continuous/discontinuous Galerkin scheme Open
A mixed continuous / discontinuous Galerkin scheme is introduced for the simulation of fluid-structure interaction problems in an isogeometric analysis framework. The properties of Non-Uniform Rational B-Spline basis functions are leverage…
View article: Large-scale Multigrid with Adaptive Galerkin Coarsening
Large-scale Multigrid with Adaptive Galerkin Coarsening Open
We propose a robust, adaptive coarse-grid correction scheme for matrix-free geometric multigrid targeting PDEs with strongly varying coefficients. The method combines uniform geometric coarsening of the underlying grid with heterogeneous c…
View article: Compact Runge-Kutta Flux Reconstruction Method for Hyperbolic Conservation Laws with Admissibility Preservation
Compact Runge-Kutta Flux Reconstruction Method for Hyperbolic Conservation Laws with Admissibility Preservation Open
Compact Runge-Kutta (cRK) Discontinuous Galerkin (DG) methods, recently introduced in [Chen, Q., Sun, Z., Xing, Y, SIAM Journal on Scientific Computing 46: A1327-A1351, 2024], are a variant of RKDG methods for solving hyperbolic conservati…
View article: On the Spectral Clustering of a Class of Multigrid Preconditioners
On the Spectral Clustering of a Class of Multigrid Preconditioners Open
This paper studies a common two-level multigrid construction for block-structured linear systems and identifies a simple way to describe how its smoothing and coarse-grid components interact. By examining the method through a collection of…
View article: On the Spectral Clustering of a Class of Multigrid Preconditioners
On the Spectral Clustering of a Class of Multigrid Preconditioners Open
This paper studies a common two-level multigrid construction for block-structured linear systems and identifies a simple way to describe how its smoothing and coarse-grid components interact. By examining the method through a collection of…
View article: A pressure-robust and parameter-free enriched Galerkin method for the Navier-Stokes equations of rotational form
A pressure-robust and parameter-free enriched Galerkin method for the Navier-Stokes equations of rotational form Open
In this paper, we develop a novel enriched Galerkin (EG) method for the steady incompressible Navier-Stokes equations in rotational form, which is both pressure-robust and parameter-free. The EG space employed here, originally proposed in …
View article: A pressure-robust and parameter-free enriched Galerkin method for the Navier-Stokes equations of rotational form
A pressure-robust and parameter-free enriched Galerkin method for the Navier-Stokes equations of rotational form Open
In this paper, we develop a novel enriched Galerkin (EG) method for the steady incompressible Navier-Stokes equations in rotational form, which is both pressure-robust and parameter-free. The EG space employed here, originally proposed in …