A mixed local discontinuous Galerkin method for a class of nonlinear problems in fluid mechanics

Authors:
Rommel Bustinza;Gabriel N. Gatica
Affiliations:
GI2MA, Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile;GI2MA, Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile
Venue:
Journal of Computational Physics
Year:
2005

Citing 9
Cited 2

Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems

SIAM Journal on Numerical Analysis
An A Priori Error Analysis of the Local Discontinuous Galerkin Method for Elliptic Problems

SIAM Journal on Numerical Analysis
Mixed hp-DGFEM for Incompressible Flows

SIAM Journal on Numerical Analysis
Thehp-local discontinuous Galerkin method for low-frequency time-harmonic Maxwell equations

Mathematics of Computation
A Posteriori Error Estimates for a Discontinuous Galerkin Approximation of Second-Order Elliptic Problems

SIAM Journal on Numerical Analysis
A Local Discontinuous Galerkin Method for Nonlinear Diffusion Problems with Mixed Boundary Conditions

SIAM Journal on Scientific Computing
An A Posteriori Error Estimate for the Local Discontinuous Galerkin Method Applied to Linear and Nonlinear Diffusion Problems

Journal of Scientific Computing
Energy norm a posteriori error estimation for mixed discontinuous Galerkin approximations of the stokes problem

Journal of Scientific Computing
Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms

Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms

A unified analysis of the local discontinuous Galerkin method for a class of nonlinear problems

Applied Numerical Mathematics - Selected papers from the first Chilean workshop on numerical analysis of partial differential equations (WONAPDE 2004)
A Posteriori Error Estimates of Discontinuous Galerkin Method for Nonmonotone Quasi-linear Elliptic Problems

Journal of Scientific Computing

Quantified Score

Hi-index	31.45

Visualization

Abstract

In this paper, we present and analyze a new mixed local discontinuous Galerkin (LDG) method for a class of nonlinear model that appears in quasi-Newtonian Stokes fluids. The approach is based on the introduction of the flux and the tensor gradient of the velocity as further unknowns. In addition, a suitable Lagrange multiplier is needed to ensure that the corresponding discrete variational formulation is well posed. This yields a two-fold saddle point operator equation as the resulting LDG mixed formulation, which is then reduced to a dual mixed formulation. Applying a nonlinear version of the well known Babuska-Brezzi theory, we prove that the discrete formulation is well posed and derive the corresponding a priori error analysis. We also develop a reliable a-posteriori error estimate and propose the associated adaptive algorithm to compute the finite element solutions. Finally, several numerical results illustrate the performance of the method and confirm its capability to localize boundary and inner layers, as well as singularities.