Smoothness-Increasing Accuracy-Conserving Filters for Discontinuous Galerkin Solutions over Unstructured Triangular Meshes

The discontinuous Galerkin (DG) method has very quickly found utility in such diverse applications as computational solid mechanics, fluid mechanics, acoustics, and electromagnetics. The DG methodology merely requires weak constraints on the fluxes between elements. This feature provides a flexibility which is difficult to match with...

Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering for Discontinuous Galerkin Solutions: Improved Errors Versus Higher-Order Accuracy

Smoothness-increasing accuracy-conserving (SIAC) filtering has demonstrated its effectiveness in raising the convergence rate of discontinuous Galerkin solutions from order k + 12 to order 2k + 1 for specific types of translation invariant meshes (Cockburn et al. in Math. Comput. 72:577–606, 2003; Curtis et al. in SIAM J. Sci. Comput. 30(1):272...

On the time-stepping stability of continuous mass-lumped and discontinuous Galerkin finite elements for the 3D acoustic wave equation

We solve the three-dimensional acoustic wave equation, discretized on tetrahedral meshes. Two methods are considered: mass-lumped continuous finite elements and the symmetric interior-penalty discontinuous Galerkin method (SIP-DG). Combining the spatial discretization with the leap-frog time-stepping scheme, which is second-order accurate and...

Smoothness-Increasing Accuracy-Conserving (SIAC) Postprocessing for Discontinuous Galerkin Solutions over Structured Triangular Meshes

Theoretically and computationally, it is possible to demonstrate that the order of accuracy of a discontinuous Galerkin (DG) solution for linear hyperbolic equations can be improved from order $k$+1 to 2$k$+1 through the use of smoothness-increasing accuracy-conserving (SIAC) filtering. However, it is a computationally complex task to perform...

Efficient Implementation of Smoothness-Increasing Accuracy-Conserving (SIAC) Filters for Discontinuous Galerkin Solutions

The discontinuous Galerkin (DG) methods provide a high-order extension of the finite volume method in much the same way as high-order or spectral/hp elements extend standard finite elements. However, lack of inter-element continuity is often contrary to the smoothness assumptions upon which many post-processing algorithms such as those used in...

Position-dependent smoothness-increasing accuracy-conserving (SIAC) filtering for improving discontinuous Galerkin solutions

Position-dependent smoothness-increasing accuracy-conserving (SIAC) filtering is a promising technique not only in improving the order of the numerical solution obtained by a discontinuous Galerkin (DG) method but also in increasing the smoothness of the field and improving the magnitude of the errors. This was initially established as an...

Local derivative post-processing: Challenges for a non-uniform mesh

Previous investigations into accuracy enhancement for the derivatives of a discontinuous Galerkin solution demonstrated that there are many ways to approach obtaining higher order accuracy in the derivatives, each with different advantageous properties. For the discontinuous Galerkin method, the order of accuracy without post-processing for the...

Goal-Oriented A-Posteriori Error Estimation and Adaptivity for Fluid-Structure Interaction

Numerical simulation of fluid-structure interaction generally requires vast computational resources. Paradoxically, the computational work is dominated by the complexity of the subsystem that is of least practical interest, viz. the fluid. The resolution of each of the many small-scale features in the fluid is prohibitively expensive. However,...

An H1(Ph)-Coercive Discontinuous Galerkin Formulation for The Poisson Problem: 1-D Analysis

Coercivity of the bilinear form in a continuum variational problem is a fundamental property for finite-element discretizations: By the classical Lax–Milgram theorem, any conforming discretization of a coercive variational problem is stable; i.e., discrete approximations are well-posed and possess unique solutions, irrespective of the specifics...

Residual based a posteriori error analysis for the advection-diffusion equation with anisotropic and discontinuous diffusion tensor

We present a residual a posteriori error estimate for the anisotropic advection diffusion equation with continuous or discontinuous diffusion tensor. Our numerical results show that the adapted mesh based on the residual is more refined in the region where anisotropy and heterogeneity effects create diffculties for the numerical solution.

Symplectic local time-stepping in non-dissipative DGTD methods applied to wave propagation problems

The Discontinuous Galerkin Time Domain (DGTD) methods are now popular for the solution of wave propagation problems required for the accurate transient modeling of systems involving electromagnetic waves in many emerging technologies. While Yees explicit, energy-conserving FDTD method is still prominent but restructed to structured grids and...

NURBS-enhanced finite element method

An improvement of the classical finite element method is proposed. It considers an exact representation of the geometry by means of the usual CAD description of the boundary with Non-Uniform Rational B-Splines (NURBS). For elements not intersecting the boundary, a standard finite element interpolation and numerical integration is used....

A Finite Element Interior Penalty Method for MHD in heterogeneous domains

The Maxwell equations in the MHD limit in heterogeneous domains composed of conducting and nonconducting regions are solved by using Lagrange finite elements and by enforcing continuities across interfaces using an interior penalty technique. The method is shown to be stable and convergent and is validated by convergence tests. It is used to...

A space-time discontinuous Galerkin finite-element discretization of the Euler equations using entropy variables

A method to numerically solve the Euler equations for fluids with general equations of state is presented. It is based on a formulation solving the conservation equations for either pressure primitive variables or entropy variables, instead of the commonly used conservation variables. We use a space-time discontinuous Galerkin finite-element...

Krylov-Subspace Preconditioners for Discontinuous Galerkin Finite Element Methods

Standard (conforming) finite element approximations of convection-dominated convection-diffusion problems often exhibit poor stability properties that manifest themselves as non-physical oscillations polluting the numerical solution. Various techniques have been proposed for the stabilisation of finite element methods (FEMs) for convection...

Flooding and drying in discontinuous Galerkin discretizations of shallow water equations

Flooding and drying in space or space-time discontinuous Galerkin (DG) discretizations provides an accurate and efficient numerical scheme. Moreover, the space-time DG method is particularly suitable for moving or deforming meshes. The shallow water equations, which can exhibit flooding and drying due to the movement of water front, are...

A high-order discontinuous Galerkin solver for 3D aerodynamic turbulent flows

This paper deals with the evaluation and validation of a recently developed parallel discontinuous Galerkin code for the numerical solution of the RANS and k-omega turbulence model equations. The main features of the code can be summarized as follows: a) high-order spatial accuracy on hybrid grids, b) fully coupled, implicit time discretization,...