In [16, 17], we constructed uniformly high order accurate discontinuous Galerkin (DG) schemes which preserve positivity of density and pressure for the Euler equations of compressible gas dynamics with the ideal gas equation of state. The technique also applies to high order accurate finite volume schemes. For the Euler equations with various source terms (e.g., gravity and chemical reactions),...

In this work, we discuss a family of finite element discretizations for the incompressible Stokes problem using continuous pressure approximations on simplicial meshes. We show that after a simple and cheap correction, the mass-fluxes obtained by the considered schemes preserve local conservation on dual cells without reducing the convergence order. This allows the direct coupling to vertex-cen...

In the last few years, two new high-order accurate methods for unstructured grids have been developed, namely the spectral volume (SV) and the spectral difference (SD) method. Both methods are related to the well-known discontinuous Galerkin (DG) method, see for instance [1], in the sense that they also use piecewise continuous polynomials as solution approximation space. The development of the...

In the last decades, Discontinuous Galerkin (DG) methods have seen rapid growth and are widely used in various application domains (see [13] for an historical introduction). This is due to their main advantage of combining the best of finite element and finite volume methods. For the time-harmonic Maxwell equations, once the problem is discretized with a DG method, finding robust solvers is a d...

The Runge–Kutta discontinuous Galerkin (RKDG) method for solving hyperbolic conservation laws is a high order finite element method, which utilizes the useful features from high resolution finite volume schemes, such as the exact or approximate Riemann solvers, TVD Runge–Kutta time discretizations, and limiters. In this paper, we investigate using the RKDG finite element method for compressible...

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...

This paper presents the application of the hybrid finite element-element free Galerkin (FE-EFG) method for the forward and inverse problems of electrical impedance tomography (EIT). The proposed method is based on the complete electrode model. Finite element (FE) and element-free Galerkin (EFG) methods are accurate numerical techniques. However, the FE technique has meshing task problems and th...

We investigate a macro-element variant of the hybridized discontinuous Galerkin (HDG) method, using patches standard simplicial elements that can have non-matching interfaces. Coupled via HDG technique, our method enables local refinement by uniform subdivision each macro-element. By enforcing one spatial discretization for all macro-elements, we arrive at problems per are embarrassingly parall...

We present an implicit large-eddy simulation of transonic buffet over the OAT15A supercritical airfoil at Mach number 0.73, angle attack 3.5 deg, and Reynolds . The is performed using a matrix-free discontinuous Galerkin (DG) method diagonally Runge–Kutta scheme on graphics processor units. propose Jacobian-free Newton–Krylov to solve nonlinear systems arising from discretization Navier–Stokes ...

We present a Ritz-Galerkin discretization on sparse grids using pre-wavelets, which allows to solve elliptic differential equations with variable coefficients for dimension d = 2, 3 and higher dimensions d > 3. The method applies multilinear finite elements. We introduce an efficient algorithm for matrix vector multiplication using a Ritz-Galerkin discretization and semi-orthogonality. This alg...