In this paper we study the temporal convergence of a locally implicit discontinuous Galerkin method for the time-domain Maxwell’s equations modeling electromagnetic waves propagation. Particularly, we wonder whether the method retains its second-order ordinary differential equation (ODE) convergence under stable simultaneous space-time grid refinement towards the true partial differential equation (PDE) solution. This is not a priori clear due to the component splitting which can introduce order...

In this paper we study the temporal convergence of a locally implicit discontinuous Galerkin method for the time-domain Maxwell’s equations modeling electromagnetic waves propagation. Particularly, we wonder whether the method retains its second-order ordinary differential equation (ODE) convergence under stable simultaneous space-time grid refinement towards the true partial differential equation (PDE) solution. This is not a priori clear due to the component splitting which can introduce order...

The initial boundary value problem for a beam is considered in the Timoshenko model. Assuming the analyticity of the initial conditions, it is proved that the problem is solvable throughout the time interval. After that, a numerical algorithm, consisting of three steps, is constructed. The solution is approximated with respect to the spatial and time variables using the Galerkin method and a Crank–Nicholson type scheme. The system of equations obtained by discretization is solved by a version of...

The initial boundary value problem for a beam is considered in the Timoshenko model. Assuming the analyticity of the initial conditions, it is proved that the problem is solvable throughout the time interval. After that, a numerical algorithm, consisting of three steps, is constructed. The solution is approximated with respect to the spatial and time variables using the Galerkin method and a Crank–Nicholson type scheme. The system of equations obtained by discretization is solved by a version...

We study the superconvergence of the finite volume method for a nonlinear elliptic problem using linear trial functions. Under the condition of $C$-uniform meshes, we first establish a superclose weak estimate for the bilinear form of the finite volume method. Then, we prove that on the mesh point set $S$, the gradient approximation possesses the superconvergence: ${max}_{P\in S}|(\nabla u-\overline{\nabla }{u}_h)\left(P\right)|=O\left(h^2\right){|lnh|}^{3/2}$, where $\overline{\nabla }$ denotes the average gradient on elements containing vertex $P$. Furthermore, by using the interpolation post-processing technique,...

A reference triangular quadratic Lagrange finite element consists of a right triangle $\widehat{K}$ with unit legs $S_1$, $S_2$, a local space $\widehat{ℒ}$ of quadratic polynomials on $\widehat{K}$ and of parameters relating the values in the vertices and midpoints of sides of $\widehat{K}$ to every function from $\widehat{ℒ}$. Any isoparametric triangular quadratic Lagrange finite element is determined by an invertible isoparametric mapping ${ℱ}_h=(F_1,F_2)\in \widehat{ℒ}\times \widehat{ℒ}$. We explicitly describe such invertible isoparametric mappings ${ℱ}_h$ for which the images ${ℱ}_h\left(S_1\right)$, ${ℱ}_h\left(S_2\right)$ of the segments $S_1$, $S_2$ are segments,...

In this paper we first study the stability of Ritz-Volterra projection (see below) and its maximum norm estimates, and then we use these results to derive some $L^{\infty }$ error estimates for finite element methods for parabolic integro-differential equations.

We give a proof of the existence of a solution of reconstruction operators used in the $P_N{P}_M$ DG schemes in one space dimension. Some properties and error estimates of the projection and reconstruction operators are presented. Then, by applying the $P_N{P}_M$ DG schemes to the linear advection equation, we study their stability obtaining maximal limits of the Courant numbers for several $P_N{P}_M$ DG schemes mostly experimentally. A numerical study explains how the stencils used in the reconstruction affect the efficiency...

We consider a viscoelastic solid in Kelvin-Voigt rheology exhibiting also plasticity with hardening and coupled with heat-transfer through dissipative heat production by viscoplastic effects and through thermal expansion and corresponding adiabatic effects. Numerical discretization of the thermodynamically consistent model is proposed by implicit time discretization, suitable regularization, and finite elements in space. Fine a-priori estimates are derived, and convergence is proved by careful successive...

We consider a viscoelastic solid in Kelvin-Voigt rheology exhibiting also plasticity with hardening and coupled with heat-transfer through dissipative heat production by viscoplastic effects and through thermal expansion and corresponding adiabatic effects. Numerical discretization of the thermodynamically consistent model is proposed by implicit time discretization, suitable regularization, and finite elements in space. Fine a-priori estimates are derived, and convergence is proved by careful...