Mahboub Baccouch

A Superconvergent Local Discontinuous Galerkin Method for Elliptic Problems

In this manuscript we investigate the convergence properties of a minimal dissipation local discontinuous Galerkin(md-LDG) method for two-dimensional diffusion problems on Cartesian meshes. Numerical computations show O(hp+1)

Asymptotically exact a posteriori LDG error estimates for one-dimensional transient convection-diffusion problems

mathcal{L}^{2}

The Local Discontinuous Galerkin Method for the Fourth-Order Euler---Bernoulli Partial Differential Equation in One Space Dimension. Part I: Superconvergence Error Analysis

convergence rates for the solution and its gradient and O(hp+2) superconvergent solutions at Radau points on enriched p-degree polynomial spaces. More precisely, a local error analysis reveals that the leading term of the LDG error for a p-degree discontinuous finite element solution is spanned by two (p+1)-degree right Radau polynomials in the x and y directions. Thus, LDG solutions are superconvergent at right Radau points obtained as a tensor product of the shifted roots of the (p+1)-degree right Radau polynomial. For tensor product polynomial spaces, the first component of the solution’s gradient is O(hp+2) superconvergent at tensor product of the roots of left Radau polynomial in x and right Radau polynomial in y while the second component is O(hp+2) superconvergent at the tensor product of the roots of the right Radau polynomial in x and left Radau polynomial in y. Several numerical simulations are performed to validate the theory.

The Local Discontinuous Galerkin Method for the Fourth-Order Euler---Bernoulli Partial Differential Equation in One Space Dimension. Part II: A Posteriori Error Estimation

In this paper, new a posteriori error estimates for the local discontinuous Galerkin (LDG) formulation applied to transient convection-diffusion problems in one space dimension are presented and analyzed. These error estimates are computationally simple and are computed by solving a local steady problem with no boundary conditions on each element. We first show that the leading error term on each element for the solution is proportional to a (p+1)-degree right Radau polynomial while the leading error term for the solutions derivative is proportional to a (p+1)-degree left Radau polynomial, when polynomials of degree at most p are used. These results are used to prove that, for smooth solutions, these error estimates at a fixed time converge to the true spatial errors in the L^2-norm under mesh refinement. More precisely, we prove that our LDG error estimates converge to the true spatial errors at O(h^p^+^5^/^4) rate. Finally, we prove that the global effectivity indices in the L^2-norm converge to unity at O(h^1^/^2) rate. Our computational results indicate that the observed numerical convergence rates are higher than the theoretical rates.

A superconvergent local discontinuous Galerkin method for the second-order wave equation on Cartesian grids

In this paper we develop and analyze a new superconvergent local discontinuous Galerkin (LDG) method for approximating solutions to the fourth-order Euler–Bernoulli beam equation in one space dimension. We prove the

Optimal energy-conserving local discontinuous Galerkin method for the one-dimensional sine-Gordon equation

L^2

Recovery-Based Error Estimator for the Discontinuous Galerkin Method for Nonlinear Scalar Conservation Laws in One Space Dimension

L2 stability of the scheme and several optimal