Discontinuous Galerkin Methods for the Biharmonic Problem on Polygonal and Polyhedral Meshes

We introduce an $hp$-version symmetric interior penalty discontinuous Galerkin finite element method (DGFEM) for the numerical approximation of the biharmonic equation on general computational meshes consisting of polygonal/polyhedral (polytopic) elements. In particular, the stability and $hp$-version a-priori error bound are derived based on the specific choice of the interior penalty parameters which allows for edges/faces degeneration. Furthermore, by deriving a new inverse inequality for a special class {of} polynomial functions (harmonic polynomials), the proposed DGFEM is proven to be stable to incorporate very general polygonal/polyhedral elements with an \emph{arbitrary} number of faces for polynomial basis with degree $p=2,3$. The key feature of the proposed method is that it employs elemental polynomial bases of total degree $\mathcal{P}_p$, defined in the physical coordinate system, without requiring the mapping from a given reference or canonical frame. A series of numerical experiments are presented to demonstrate the performance of the proposed DGFEM on general polygonal/polyhedral meshes

Fully computable a posteriori error bounds for eigenfunctions

Fully computable a posteriori error estimates for eigenfunctions of compact self-adjoint operators in Hilbert spaces are derived. The problem of ill-conditioning of eigenfunctions in case of tight clusters and multiple eigenvalues is solved by estimating the directed distance between the spaces of exact and approximate eigenfunctions. Derived upper bounds apply to various types of eigenvalue problems, e.g. to the (generalized) matrix, Laplace, and Steklov eigenvalue problems. These bounds are suitable for arbitrary conforming approximations of eigenfunctions, and they are fully computable in terms of approximate eigenfunctions and two-sided bounds of eigenvalues. Numerical examples illustrate the efficiency of the derived error bounds for eigenfunctions.Comment: 27 pages, 8 tables, 9 figure

The a posteriori error estimates of the Ciarlet-Raviart mixed finite element method for the biharmonic eigenvalue problem

The biharmonic equation/eigenvalue problem is one of the fundamental model problems in mathematics and physics and has wide applications. In this paper, for the biharmonic eigenvalue problem, based on the work of Gudi [Numer. Methods Partial Differ. Equ., 27 (2011), 315-328], we study the a posteriori error estimates of the approximate eigenpairs obtained by the Ciarlet-Raviart mixed finite element method. We prove the reliability and efficiency of the error estimator of the approximate eigenfunction and analyze the reliability of the error estimator of the approximate eigenvalues. We also implement the adaptive calculation and exhibit the numerical experiments which show that our method is efficient and can get an approximate solution with high accuracy

Computational Engineering

The focus of this Computational Engineering Workshop was on the mathematical foundation of state-of-the-art and emerging finite element methods in engineering analysis. The 52 participants included mathematicians and engineers with shared interest on discontinuous Galerkin or Petrov-Galerkin methods and other generalized nonconforming or mixed finite element methods

Local parameter selection in the C0C^0 interior penalty method for the biharmonic equation

The symmetric $C^0$ interior penalty method is one of the most popular discontinuous Galerkin methods for the biharmonic equation. This paper introduces an automatic local selection of the involved stability parameter in terms of the geometry of the underlying triangulation for arbitrary polynomial degrees. The proposed choice ensures a stable discretization with guaranteed discrete ellipticity constant. Numerical evidence for uniform and adaptive mesh-refinement and various polynomial degrees supports the reliability and efficiency of the local parameter selection and recommends this in practice. The approach is documented in 2D for triangles, but the methodology behind can be generalized to higher dimensions, to non-uniform polynomial degrees, and to rectangular discretizations. Two appendices present the realization of our proposed parameter selection in various established finite element software packages as well as a detailed documentation of a self-contained MATLAB program for the lowest-order $C^0$ interior penalty method