Abstract
This paper presents for the first time the derivation of an hp a posteriori error estimator for the symmetric interior penalty discontinuous Galerkin finite element method for linear elastic analysis. Any combination of Neumann and Dirichlet boundary conditions are admissible in the formulation, including applying Neumann and Dirichlet on different components on the same region of the boundary. Therefore, the error estimator is applicable to a variety of physical problems. The error estimator is incorporated into an hp-adaptive finite element solver and verified against smooth and non-smooth problems with closed-form analytical solutions, as well as, being demonstrated on a non-smooth problem with complex boundary conditions. The hp-adaptive finite element analyses achieve exponential rates of convergence. The performances of the hp-adaptive scheme are contrasted against uniform and adaptive h refinement. This paper provides a complete framework for adaptivity in the symmetric interior penalty discontinuous Galerkin finite element method for linear elastic analysis.
Highlights
The discretization of partial differential equations for numerical computation facilitates the solution of physical systems, it introduces approximation errors
We introduce the continuous weak formulation of problem (1), this formulation is only used in the analysis for the error estimator and it is not implemented in the code
We introduce the polynomial degrees for the approximation in our Discontinuous Galerkin (DG) method
Summary
The discretization of partial differential equations for numerical computation facilitates the solution of physical systems, it introduces approximation errors. We present a new error estimator for linear elasticity based on a Discontinuous Galerkin (DG) finite element method. DG methods were introduced in the early 1970s as a way to numerically solve first-order hyperbolic problems [2]. A variety of DG methods have been developed in the decades [6], among them is the subclass of interior penalty DG methods which are stable and, in the authors’ opinion, easy to implement. In [7] the first residual-based energy-norm error estimator for the symmetric interior penalty discontinuous Galerkin finite element method is presented. The symmetric interior penalty discontinuous Galerkin finite element method is introduced, and Section 5 provides the a priori convergence results of the method.
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