Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems II: Strongly monotone quasi-Newtonian flows

In this article we develop both the a priori and a posteriori error analysis of hp–version interior penalty discontinuous Galerkin finite element methods for strongly monotone quasi-Newtonian fluid flows in a bounded Lipschitz domain $\Omega \subset R^{d}, d$ = 2,3. In the latter case, computable upper and lower bounds on the error are derived in terms of a natural energy norm which are explicit in the local mesh size and local polynomial degree of the approximating finite element method. A series of numerical experiments illustrate the performance of the proposed a posteriori error indicators within an automatic hp–adaptive refinement algorithm

Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems II: strongly monotone quasi-Newtonian flows

In this article we develop both the a priori and a posteriori error analysis of hp– version interior penalty discontinuous Galerkin finite element methods for strongly monotone quasi-Newtonian fluid flows in a bounded Lipschitz domain Ω ⊂ R^d, d = 2, 3. In the latter case, computable upper and lower bounds on the error are derived in terms of a natural energy norm which are explicit in the local mesh size and local polynomial degree of the approximating finite element method. A series of numerical experiments illustrate the performance of the proposed a posteriori error indicators within an automatic hp–adaptive refinement algorithm

Nonoverlapping domain decomposition preconditioners for discontinuous Galerkin finite element methods in H2H^2-type norms

We analyse the spectral bounds of nonoverlapping domain decomposition preconditioners for $hp$-version discontinuous Galerkin finite element methods in $H^2$-type norms, which arise in applications to fully nonlinear Hamilton--Jacobi--Bellman partial differential equations. We show that for a symmetric model problem, the condition number of the preconditioned system is at most of order $1+ p^6 H^3 /q^3 h^3$, where $H$ and $h$ are respectively the coarse and fine mesh sizes, and $q$ and $p$ are respectively the coarse and fine mesh polynomial degrees. This represents the first result for this class of methods that explicitly accounts for the dependence of the condition number on $q$, and its sharpness is shown numerically. The key analytical tool is an original optimal order approximation result between fine and coarse discontinuous finite element spaces.\ud \ud We then go beyond the model problem and show computationally that these methods lead to efficient and competitive solvers in practical applications to nonsymmetric, fully nonlinear Hamilton--Jacobi--Bellman equations

Locking‐free DGFEM for elasticity problems in polygons

The h‐version of the discontinuous Galerkin finite element method (h‐DGFEM) for nearly incompressible linear elasticity problems in polygons is analysed. It is proved that the scheme is robust (locking‐free) with respect to volume locking, even in the absence of H2‐regularity of the solution. Furthermore, it is shown that an appropriate choice of the finite element meshes leads to robust and optimal algebraic convergence rates of the DGFEM even if the exact solutions do not belong to H

A locking-free discontinuous Galerkin method for linear elastic Steklov eigenvalue problem

In this paper, a discontinuous Galerkin finite element method of Nitsche's version for the Steklov eigenvalue problem in linear elasticity is presented. The a priori error estimates are analyzed under a low regularity condition, and the robustness with respect to nearly incompressible materials (locking-free) is proven. Furthermore, some numerical experiments are reported to show the effectiveness and robustness of the proposed method.Comment: 25 pages, 6 figure

A Posteriori Error Analysis of hp-Version Discontinuous Galerkin Finite Element Methods for Second-Order Quasilinear Elliptic Problems

We develop the a-posteriori error analysis of hp-version interior-penalty discontinuous Galerkin finite element methods for a class of second-order quasilinear elliptic partial differential equations. Computable upper and lower bounds on the error are derived in terms of a natural (mesh-dependent) energy norm. The bounds are explicit in the local mesh size and the local degree of the approximating polynomial. The performance of the proposed estimators within an automatic hp-adaptive refinement procedure is studied through numerical experiments

Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems II: strongly monotone quasi-Newtonian flows

In this article, we develop the a priori and a posteriori error analysis of hp-version interior penalty discontinuous Galerkin finite element methods for strongly monotone quasi-Newtonian fluid flows in a bounded Lipschitz domain Ω ⊂ ℝd, d = 2, 3. In the latter case, computable upper and lower bounds on the error are derived in terms of a natural energy norm, which are explicit in the local mesh size and local polynomial degree of the approximating finite element method. A series of numerical experiments illustrate the performance of the proposed a posteriori error indicators within an automatic hp-adaptive refinement algorith

Review of discontinuous Galerkin finite element methods for partial differential equations on complicated domains

The numerical approximation of partial differential equations (PDEs) posed on complicated geometries, which include a large number of small geometrical features or microstructures, represents a challenging computational problem. Indeed, the use of standard mesh generators, employing simplices or tensor product elements, for example, naturally leads to very fine finite element meshes, and hence the computational effort required to numerically approximate the underlying PDE problem may be prohibitively expensive. As an alternative approach, in this article we present a review of composite/agglomerated discontinuous Galerkin finite element methods (DGFEMs) which employ general polytopic elements. Here, the elements are typically constructed as the union of standard element shapes; in this way, the minimal dimension of the underlying composite finite element space is independent of the number of geometrical features. In particular, we provide an overview of hp-version inverse estimates and approximation results for general polytopic elements, which are sharp with respect to element facet degeneration. On the basis of these results, a priori error bounds for the hp-DGFEM approximation of both second-order elliptic and first-order hyperbolic PDEs will be derived. Finally, we present numerical experiments which highlight the practical application of DGFEMs on meshes consisting of general polytopic elements

Nonstandard Finite Element Methods

[no abstract available