The formulation and the implementation of boundary conditions within the context of the quadrature-free form of the discontinuous Galerkin method are presented for several types of boundary conditions for the Euler equations. An important feature of the discontinuous Galerkin method is that the interior point algorithm is well behaved in the neighborhood of the boundary and requires no modi cat...

This paper concerns with numerical approximations of solutions of fully nonlinear second order partial differential equations (PDEs). A new notion of weak solutions, called moment solutions, is introduced for fully nonlinear second order PDEs. Unlike viscosity solutions, moment solutions are defined by a constructive method, called the vanishing moment method, and hence, they can be readily com...

Abstract A spectral‐Galerkin method based on Legendre‐Fourier approximation for fourth‐order problems in cylindrical regions is studied this paper. By the coordinate transformation, a three‐dimensional problem region transformed into sequence of decoupled with two dimensions and corresponding pole conditions are also derived. With appropriately constructed weighted Sobolev space, weak form esta...

A finite element method, called generalized Weak Galerkin (gWG), is introduced for the Stokes equation by using a new weak gradient notion in usual approach. The gWG method allows polynomial elements of arbitrary order any combination velocity variable, but with proper constraints on pressure due to inf-sup condition. Error estimates are derived approximation mesh-dependent energy norm, as well...

High-order triangle-based discontinuous Galerkin (DG) methods for hyperbolic equations on a rotating sphere are presented. The DG method can be characterized as the fusion of finite elements with finite volumes. This DG formulation uses high-order Lagrange polynomials on the triangle using nodal sets up to 15th order. The finite element-type area integrals are evaluated using order 2N Gauss cub...

A novel formulation is given for the equations of viscous incompressible magnetohydrodynamics with realistic (“nonideal”) boundary conditions that account for the fluid’s interaction with the outside world. The global-in-time existence of weak solutions is established via the Galerkin method. AMS (MOS) Subject Classification. 76W05, 35Q30, 35Q60, 65M60

The purpose of this paper is to show that the Fourier-based Nonlinear Galerkin Method (NLGM) constructs suitable weak solutions to the periodic Navier–Stokes equations in three dimensions. We re-interpret NLGM as a Large-Eddy Simulation technique (LES) and we rigorously deduce a relationship between the mesh size and the large-eddy scale.

Three kinds of fragile points methods based on Petrov-Galerkin weak-forms (PG-FPMs) are proposed for analyzing heat conduction problems in nonhomogeneous anisotropic media. This is a follow-up the previous study original FPM symmetric Galerkin weak-form. The trial function piecewise-continuous, written as local Taylor expansions at points. A modified radial basis function-based differential qua...

The convergence of the discontinuous Galerkin method for the nonlinear (cubic) Schrödinger equation is analyzed in this paper. We show the existence of the resulting approximations and prove optimal order error estimates in L∞(L2). These estimates are valid under weak restrictions on the space-time mesh.