This article is devoted to the discretization of source terms and boundary conditions using discontinuous Galerkin schemes with an arbitrary high order of accuracy in space and time for the solution of hyperbolic conservation laws on unstructured triangular meshes. The building block of the method is a particular numerical flux function at the element interfaces based on the solution of Generalized Riemann Problems (GRPs) with piecewise polynomial initial data. The solution of the generalized Riemann...

In this paper, we present some interesting connections between a number of Riemann-solver free approaches to the numerical solution of multi-dimensional systems of conservation laws. As a main part, we present a new and elementary derivation of Fey’s Method of Transport (MoT) (respectively the second author’s ICE version of the scheme) and the state decompositions which form the basis of it. The only tools that we use are quadrature rules applied to the moment integral used in the gas kinetic derivation...

In this paper, we present some interesting connections between a number of Riemann-solver free approaches to the numerical solution of multi-dimensional systems of conservation laws. As a main part, we present a new and elementary derivation of Fey's Method of Transport (MoT) (respectively the second author's ICE version of the scheme) and the state decompositions which form the basis of it. The only tools that we use are quadrature rules applied to the moment integral used in the...

We discretize the nonlinear Schrödinger equation, with Dirichlet boundary conditions, by a linearly implicit two-step finite element method which conserves the $L^2$ norm. We prove optimal order a priori error estimates in the $L^2$ and $H^1$ norms, under mild mesh conditions for two and three space dimensions.

We discretize the nonlinear Schrödinger equation, with Dirichlet boundary conditions, by a linearly implicit two-step finite element method which conserves the L2 norm. We prove optimal order a priori error estimates in the L2 and H1 norms, under mild mesh conditions for two and three space dimensions.

The backward Euler algorithm for the multidimensional nonhomogeneous heat equation is analyzed, based on the finite element method. The existence and uniqueness of the numerical solution is investigated. Also, the convergence of the numerical solutions is studied.

In this paper we mathematically analyse an evolution variational inequality which formulates the double critical-state model for type-II superconductivity in 3D space and propose a finite element method to discretize the formulation. The double critical-state model originally proposed by Clem and Perez-Gonzalez is formulated as a model in 3D space which characterizes the nonlinear relation between the electric field, the electric current, the perpendicular component of the electric current...

Let $\Omega$ be a bounded domain in ${ℝ}^n$ with a smooth boundary $\Gamma$. In this work we study the existence of solutions for the following boundary value problem: $$\frac{{\partial }^{2}y}{\partial t^2}-M\left({\int }_{\Omega }{\left|\nabla y\right|}^2\mathrm{d}x\right)\Delta y-\frac{\partial }{\partial t}\Delta y=f\left(y\right)\text{in}Q=\Omega \times (0,\infty ),.1y=0\text{in}{\Sigma }_1={\Gamma }_1\times (0,\infty ),M\left({\int }_{\Omega }{\left|\nabla y\right|}^2\mathrm{d}x\right)\frac{\partial y}{\partial \nu }+\frac{\partial }{\partial t}\left(\frac{\partial y}{\partial \nu }\right)=g\text{in}{\Sigma }_0={\Gamma }_0\times (0,\infty ),y\left(0\right)=y_0,\frac{\partial y}{\partial t}\left(0\right)=y_1\text{in}\Omega ,\left(1\right)$$ where $M$ is a $C^1$-function such that $M\left(\lambda \right)\ge {\lambda }_0>0$ for every $\lambda \ge 0$ and $f\left(y\right)={\left|y\right|}^{\alpha }y$ for $\alpha \ge 0$.

Karátson and Korotov developed a sharp upper global a posteriori error estimator for a large class of nonlinear problems of elliptic type, see J. Karátson, S. Korotov (2009). The goal of this paper is to check its numerical performance, and to demonstrate the efficiency and accuracy of this estimator on the base of quasilinear elliptic equations of the second order. The focus will be on the technical and numerical aspects and on the components of the error estimation, especially on the adequate...

In this paper, we deal with the optimal choice of the parameter $\gamma$ for augmented Lagrangian preconditioning of GMRES method for efficient solution of linear systems obtained from discretization of the incompressible Navier-Stokes equations. We consider discretization of the equations using the B-spline based isogeometric analysis approach. We are interested in the dependence of the convergence on the parameter $\gamma$ for various problem parameters (Reynolds number, mesh refinement) and especially for...

The paper is concerned with the analysis of the space-time discontinuous Galerkin method (STDGM) applied to the numerical solution of the nonstationary nonlinear convection-diffusion initial-boundary value problem in a time-dependent domain formulated with the aid of the arbitrary Lagrangian-Eulerian (ALE) method. In the formulation of the numerical scheme we use the nonsymmetric, symmetric and incomplete versions of the space discretization of diffusion terms and interior and boundary penalty....

We consider the coupling between three-dimensional (3D) and one-dimensional (1D) fluid-structure interaction (FSI) models describing blood flow inside compliant vessels. The 1D model is a hyperbolic system of partial differential equations. The 3D model consists of the Navier-Stokes equations for incompressible Newtonian fluids coupled with a model for the vessel wall dynamics. A non standard formulation for the Navier-Stokes equations is adopted to have suitable boundary conditions for the...

We consider the original DG method for solving the advection-reaction equations with arbitrary velocity in $d$ space dimensions. For triangulations satisfying the flow condition, we first prove that the optimal convergence rate is of order $k+1$ in the $L_2$-norm if the method uses polynomials of order $k$. Then, a very simple derivative recovery formula is given to produce an approximation to the derivative in the flow direction which superconverges with order $k+1$. Further we consider a residual-based a posteriori...

The present paper studies an optimization problem of dynamically loaded cylindrical tubes. This is a problem of linear elasticity theory. As we search for the optimal thickness of the tube which minimizes the displacement under forces, this is a problem of shape optimization. The mathematical model is given by a differential equation (ODE and PDE, respectively); the mechanical problem is described as an optimal control problem. We consider both the stationary (time independent) and the transient...