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

A Nonconforming Finite Element Approximation for the von Karman Equations

In this paper, a nonconforming finite element method has been proposed and analyzed for the von Karman equations that describe bending of thin elastic plates. Optimal order error estimates in broken energy and $H^1$ norms are derived under minimal regularity assumptions. Numerical results that justify the theoretical results are presented.Comment: The paper is submitted to an international journa

Nonoverlapping domain decomposition preconditioners for discontinuous Galerkin approximations of Hamilton--Jacobi--Bellman equations

We analyse a class of nonoverlapping domain decomposition preconditioners for nonsymmetric linear systems arising from discontinuous Galerkin finite element approximation of fully nonlinear Hamilton--Jacobi--Bellman (HJB) partial differential equations. These nonsymmetric linear systems are uniformly bounded and coercive with respect to a related symmetric bilinear form, that is associated to a matrix $\mathbf{A}$. In this work, we construct a nonoverlapping domain decomposition preconditioner $\mathbf{P}$, that is based on $\mathbf{A}$, and we then show that the effectiveness of the preconditioner for solving the} nonsymmetric problems can be studied in terms of the condition number $\kappa(\mathbf{P}^{-1}\mathbf{A})$. In particular, we establish the bound $\kappa(\mathbf{P}^{-1}\mathbf{A}) \lesssim 1+ p^6 H^3 /q^3 h^3$, 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 such result for this class of methods that explicitly accounts for the dependence of the condition number on $q$; our analysis is founded upon an original optimal order approximation result between fine and coarse discontinuous finite element spaces. Numerical experiments demonstrate the sharpness of this bound. Although the preconditioners are not robust with respect to the polynomial degree, our bounds quantify the effect of the coarse and fine space polynomial degrees. Furthermore, we show computationally that these methods are effective in practical applications to nonsymmetric, fully nonlinear HJB equations under $h$-refinement for moderate polynomial degrees

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

Adaptive discontinuous Galerkin approximations to fourth order parabolic problems

An adaptive algorithm, based on residual type a posteriori indicators of errors measured in $L^{\infty}(L^2)$ and $L^2(L^2)$ norms, for a numerical scheme consisting of implicit Euler method in time and discontinuous Galerkin method in space for linear parabolic fourth order problems is presented. The a posteriori analysis is performed for convex domains in two and three space dimensions for local spatial polynomial degrees $r\ge 2$. The a posteriori estimates are then used within an adaptive algorithm, highlighting their relevance in practical computations, which results into substantial reduction of computational effort

Error estimates for the numerical approximation of a distributed optimal control problem governed by the von K\'arm\'an equations

In this paper, we discuss the numerical approximation of a distributed optimal control problem governed by the von Karman equations, defined in polygonal domains with point-wise control constraints. Conforming finite elements are employed to discretize the state and adjoint variables. The control is discretized using piece-wise constant approximations. A priori error estimates are derived for the state, adjoint and control variables under minimal regularity assumptions on the exact solution. Numerical results that justify the theoretical results are presented