Slimane Adjerid

A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems

Abstract We analyze the spatial discretization errors associated with solutions of one-dimensional hyperbolic conservation laws by discontinuous Galerkin methods (DGMs) in space. We show that the leading term of the spatial discretization error with piecewise polynomial approximations of degree p is proportional to a Radau polynomial of degree p +1 on each element. We also prove that the local and global discretization errors are O( Δx 2( p +1) ) and O( Δx 2 p +1 ) at the downwind point of each element. This strong superconvergence enables us to show that local and global discretization errors converge as O( Δx p +2 ) at the remaining roots of Radau polynomial of degree p +1 on each element. Convergence of local and global discretization errors to the Radau polynomial of degree p +1 also holds for smooth solutions as p →∞. These results are used to construct asymptotically correct a posteriori estimates of spatial discretization errors that are effective for linear and nonlinear conservation laws in regions where solutions are smooth.

A moving finite element method with error estimation and refinement for one-dimensional time dependent partial differential equations

We discuss a moving finite element method for solving vector systems of time dependent partial differential equations in one space dimension. The mesh is moved so as to equidistribute the spatial component of the discretization error in

A posteriori discontinuous finite element error estimation for two-dimensional hyperbolic problems

H^1

A posteriori error estimation with finite element methods of lines for one-dimensional parabolic systems

. We present a method of estimating this error by using p-hierarchic finite elements. The error estimate is also used in an adaptive mesh refinement procedure to give an algorithm that combines mesh movement and refinement.We discretize the partial differential equations in space using a Galerkin procedure with piecewise linear elements to approximate the solution and quadratic elements to estimate the error. A system of ordinary differential equations for mesh velocities are used to control element motions. We use existing software for stiff ordinary differential equations for the temporal integration of the solution, the error estimate, and the mesh motion. Computational results using a code based on our method are presented for several examples.

A POSTERIORI ERROR ESTIMATION FOR THE FINITE ELEMENT METHOD-OF-LINES SOLUTION OF PARABOLIC PROBLEMS

Abstract We analyze the discontinuous finite element errors associated with p-degree solutions for two-dimensional first-order hyperbolic problems. We show that the error on each element can be split into a dominant and less dominant component and that the leading part is O(hp+1) and is spanned by two (p+1)-degree Radau polynomials in the x and y directions, respectively. We show that the p-degree discontinuous finite element solution is superconvergent at Radau points obtained as a tensor product of the roots of (p+1)-degree Radau polynomial. For a linear model problem, the p-degree discontinuous Galerkin solution flux exhibits a strong O(h2p+2) local superconvergence on average at the element outflow boundary. We further establish an O(h2p+1) global superconvergence for the solution flux at the outflow boundary of the domain. These results are used to construct simple, efficient and asymptotically correct a posteriori finite element error estimates for multi-dimensional first-order hyperbolic problems in regions where solutions are smooth.

The Discontinuous Galerkin Method for Two-Dimensional Hyperbolic Problems. Part I: Superconvergence Error Analysis

SummaryConsider the solution of one-dimensional linear initial-boundary value problems by a finite element method of lines using a piecewisePth-degree polynomial basis. A posteriori estimates of the discretization error are obtained as the solutions of either local parabolic or local elliptic finite element problems using piecewise polynomial corrections of degreep+1 that vanish at element ends. Error estimates computed in this manner are shown to converge in energy under mesh refinement to the exact finite element discretization error. Computational results indicate that the error estimates are robust over a wide range of mesh spacings and polynomial degrees and are, furthermore, applicable in situations that are not supported by the analysis.

Superconvergence of discontinuous finite element solutions for transient convection-diffusion problems

Babuska and Yu constructed a posteriori estimates for finite element discretization errors of linear elliptic problems utilizing a dichotomy principal stating that the errors of odd-order approximations arise near element edges as mesh spacing decreases while those of even-order approximations arise in element interiors. We construct similar a posteriori estimates for the spatial errors of finite element method-of-lines solutions of linear parabolic partial differential equations on square-element meshes. Error estimates computed in this manner are proven to be asymptotically correct; thus, they converge in strain energy under mesh refinement at the same rate as the actual errors.

High-order finite element methods for singularly perturbed elliptic and parabolic problems

Abstract In this paper we investigate the superconvergence properties of the discontinuous Galerkin method applied to scalar first-order hyperbolic partial differential equations on triangular meshes. We show that the discontinuous finite element solution is O(hp+2) superconvergent at the Legendre points on the outflow edge for triangles having one outflow edge. For triangles having two outflow edges the finite element error is O(hp+2) superconvergent at the end points of the inflow edge. Several numerical simulations are performed to validate the theory. In Part II of this work we explicitly write down a basis for the leading term of the error and construct asymptotically correct a posteriori error estimates by solving local hyperbolic problems with no boundary conditions on more general meshes.

A Local Refinement Finite-Element Method for Two-Dimensional Parabolic Systems

We present a study of the local discontinuous Galerkin method for transient convection–diffusion problems in one dimension. We show that p-degree piecewise polynomial discontinuous finite element solutions of convection-dominated problems are O(Δxp+2) superconvergent at Radau points. For diffusion- dominated problems, the solution’s derivative is O(Δxp+2) superconvergent at the roots of the derivative of Radau polynomial of degree p+1. Using these results, we construct several asymptotically exact a posteriori finite element error estimates. Computational results reveal that the error estimates are asymptotically exact.

Hierarchical finite element bases for triangular and tetrahedral elements

We develop a framework for applying high-order finite element methods to singularly-perturbed elliptic and parabolic differential systems that utilizes special quadrature rules to confine spurious effects, such as excess diffusion and nonphysical oscillations, to boundary and interior layers. This approach is more suited for use with adaptive mesh-refinement and order-variation techniques than other problem-dependent methods. Quadrature rules, developed for two-point convection-diffusion and reaction-diffusion problems, are used with finite element software to solve examples involving ordinary and partial differential equations. Numerical artifacts are confined to layers for all combinations of meshes, orders and singular perturbation parameters that were tested. Radau or Lobatto quadrature used with the finite element method to solve, respectively, convection-and reaction-diffusion problems provide the benefits of the specialized quadrature formulas and are simpler to implement.