The paper presents a Galerkin approach for the solution of nonlinear beam equations. The approach is energy consistent, i.e., it is shown that the weighted residual integral describes energy flow. The Galerkin approach gives accurate results with less degrees of freedom as compared to low-order finite element formulation. The Galerkin approach also leads to a nonlinear order-reduction technique...

A linearized backward Euler Galerkin-mixed finite element method is investigated for the time-dependent Ginzburg–Landau (TDGL) equations under the Lorentz gauge. By introducing the induced magnetic field σ = curlA as a new variable, the Galerkin-mixed FE scheme offers many advantages over conventional Lagrange type Galerkin FEMs. An optimal error estimate for the linearized Galerkin-mixed FE sc...

A new Petrov-Galerkin approach for dealing with sharp or abrupt field changes in Discontinuous Galerkin (DG) reduced order modelling (ROM) is outlined in this paper. This method presents a natural and easy way to introduce a diffusion term into ROM without tuning/optimising and provides appropriate modeling and stablisation for the numerical solution of high order nonlinear PDEs. The approach i...

In this paper we investigate the relationship between the continuous and the discontinuous Galerkin methods for elliptic problems. In particular, we show that the continuous Galerkin method can be interpreted as the limit of a discontinuous Galerkin method when a stabilization parameter tends to innnity. Based on this observation we derive a method for computing a conservative approximation of ...

Four primal discontinuous Galerkin methods are applied to solve reactive transport problems, namely, Oden-Babuška-Baumann DG (OBB-DG), non-symmetric interior penalty Galerkin (NIPG), symmetric interior penalty Galerkin (SIPG), and incomplete interior penalty Galerkin (IIPG). A unified a posteriori residual-type error estimation is derived explicitly for these methods. From the computed solution...

Over the last few years, several mixed discontinuous Galerkin finite element methods (DGFEM) have been proposed for the discretization of incompressible fluid flow problems. We mention here only the piecewise solenoidal discontinuous Galerkin methods introduced in [5,25], the local discontinuous Galerkin methods of [12,11], and the interior penalty methods studied in [24,33,18]. Some of the mai...

We study a multiscale discontinuous Galerkin method introduced in [10] that reduces the computational complexity of the discontinuous Galerkin method, seemingly without adversely affecting the quality of results. For a stabilized variant we are able to obtain the same error estimates for the advection-diffusion equation as for the usual discontinuous Galerkin method. We assess the stability of ...

A series of efforts were made to solve a simple ablation problem with gas motion through the porous media employing finite element based Galerkin and Discontinuous Galerkin methods. First, one-dimensional solutions of Euler and magneto-hydrodynamics (MHD) equations are presented for comparison with analytical results, to validate the code. The spurious oscillations of standard Galerkin approach...

In this paper we introduce a new method for solving partial and ordinary differential equations with large first, second and third derivatives of the solution in some part of the domain using the finite element technique (here called the Galerkin-Gokhman method). The method is based on the application of the Galerkin method to a modified differential equation. The exact solution of the modified...

We introduce a new and efficient Chebyshev-Legendre Galerkin method for elliptic problems. The new method is based on a Legendre-Galerkin formulation, but only the Chebyshev-Gauss-Lobatto points are used in the computation. Hence, it enjoys advantages of both the LegendreGalerkin and Chebyshev-Galerkin methods.