Related papers: High order discontinuous Galerkin methods on surfa…
We propose and analyze a new hybridizable discontinuous Galerkin (HDG) method for second-order elliptic problems. Our method is obtained by inserting the $L^2$-orthogonal projection onto the approximate space for a numerical trace into all…
We propose efficient and parallel algorithms for the implementation of the high-order continuous time Galerkin method for dissipative and wave propagation problems. By using Legendre polynomials as shape functions, we obtain a special…
An integro-differential equation, modeling dynamic fractional order viscoelasticity, with a Mittag-Leffler type convolution kernel is considered. A discontinuous Galerkin method, based on piecewise constant polynomials is formulated for…
We present a high order scheme for approximating kinetic equations with stiff relaxation. The objective is to provide efficient methods for solving the underlying system of conservation laws. The construction is based on several…
We propose an discontinuous Galerkin local orthogonal decomposition multiscale method for convection-diffusion problems with rough, heterogeneous, and highly varying coefficients. The properties of the multiscale method and the…
In this paper, we analyze a family of hybridizable discontinuous Galerkin (HDG) methods for second order elliptic problems in two and three dimensions. The methods use piecewise polynomials of degree $k\geqslant 0$ for both the flux and…
An adaptive algorithm, based on residual type a posteriori indicators of errors measured in $L^{\infty}(L^2)$ and $L^2(L^2)$ norms, for a numerical scheme consisting of implicit Euler method in time and discontinuous Galerkin method in…
A conforming discontinuous Galerkin finite element method is introduced for solving the biharmonic equation. This method, by its name, uses discontinuous approximations and keeps simple formulation of the conforming finite element method at…
We consider a system of second order non-linear elliptic partial differential equations that models the equilibrium configurations of a two dimensional planar bistable nematic liquid crystal device. Discontinuous Galerkin finite element…
We propose energy-conserving discontinuous Galerkin (DG) methods for symmetric linear hyperbolic systems on general unstructured meshes. Optimal a priori error estimates of order $k+1$ are obtained for the semi-discrete scheme in one…
We introduce a new hybridized discontinuous Galerkin method for the incompressible magnetohydrodynamics equations. If particular velocity, pressure, magnetic field, and magnetic pressure spaces are employed for both element and trace…
A new high order accurate semi-implicit space-time Discontinuous Galerkin method on staggered grids, for the simulation of viscous incompressible flows on two-dimensional domains is presented. The designed scheme is of the Arbitrary…
Based on the Jacobi polynomial expansion, an arbitrary high-order Discontinuous Galerkin solver for compressible flows on unstructured meshes is proposed in the present work. First, we construct orthogonal polynomials for 2D and 3D…
We consider discontinuous Galerkin methods for an elliptic distributed optimal control problem and we propose multigrid methods to solve the discretized system. We prove that the $W$-cycle algorithm is uniformly convergent in the energy…
This paper is dedicated to the development of numerical analysis for high-order methods solving partial differential equations on scattered point clouds. We build a novel geometric error analysis framework by estimating the error in the…
We present an arbitrary order discontinuous Galerkin finite element method for solving the fourth-order curl problem using a reconstructed discontinuous approximation method. It is based on an arbitrarily high-order approximation space with…
Many differential equations with physical backgrounds are described as gradient systems, which are evolution equations driven by the gradient of some functionals, and such problems have energy conservation or dissipation properties. For…
We present analysis of two lowest-order hybridizable discontinuous Galerkin methods for the Stokes problem, while making only minimal regularity assumptions on the exact solution. The methods under consideration have previously been shown…
We present a high-order hybridizable discontinuous Galerkin method for the numerical solution of time-dependent three-phase flow in heterogeneous porous media. The underlying algorithm is a semi-implicit operator splitting approach that…
We consider a discontinuous Galerkin method for the numerical solution of boundary value problems in two-dimensional domains with curved boundaries. A key challenge in this setting is the potential loss of convergence order due to…