2,033 research outputs found
Local Discontinuous Galerkin Finite Element Method and Error Estimates for One Class of Sobolev Equation
In this paper we present a numerical scheme based on the local discontinuous Galerkin (LDG) finite element method for one class of Sobolev equations, for example, generalized equal width Burgers equation. The proposed scheme will be proved to have good numerical stability and high order accuracy for arbitrary nonlinear convection flux, when time variable is continuous. Also an optimal error estimate is obtained for the fully discrete scheme, when time is discreted by the second order explicit total variation diminishing (TVD) Runge-Kutta time-marching. Finally some numerical results are given to verify our analysis for the scheme
Optimal Error Estimates for the hp–Version Interior Penalty Discontinuous Galerkin Finite Element Method
We consider the hp-version interior penalty discontinuous Galerkin finite element method (hp-DGFEM) for second-order linear reaction-diffusion equations. To the best of our knowledge, the sharpest known error bounds for the hp-DGFEM are due to Riviere, Wheeler and Girault [9] and due to Houston, Schwab and Süli [6] which are optimal with respect to the meshsize h but suboptimal with respect to the polynomial degree p by half an order of p. We present improved error bounds in the energy norm, by introducing a new function space framework. More specifically, assuming that the solutions belong element-wise to an augmented Sobolev space, we deduce hp-optimal error bounds
Analysis of a space--time hybridizable discontinuous Galerkin method for the advection--diffusion problem on time-dependent domains
This paper presents the first analysis of a space--time hybridizable
discontinuous Galerkin method for the advection--diffusion problem on
time-dependent domains. The analysis is based on non-standard local trace and
inverse inequalities that are anisotropic in the spatial and time steps. We
prove well-posedness of the discrete problem and provide a priori error
estimates in a mesh-dependent norm. Convergence theory is validated by a
numerical example solving the advection--diffusion problem on a time-dependent
domain for approximations of various polynomial degree
Robust error estimates in weak norms for advection dominated transport problems with rough data
We consider mixing problems in the form of transient convection--diffusion
equations with a velocity vector field with multiscale character and rough
data. We assume that the velocity field has two scales, a coarse scale with
slow spatial variation, which is responsible for advective transport and a fine
scale with small amplitude that contributes to the mixing. For this problem we
consider the estimation of filtered error quantities for solutions computed
using a finite element method with symmetric stabilization. A posteriori error
estimates and a priori error estimates are derived using the multiscale
decomposition of the advective velocity to improve stability. All estimates are
independent both of the P\'eclet number and of the regularity of the exact
solution
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