Stable splitting of polyharmonic operators by generalized Stokes systems

  title={Stable splitting of polyharmonic operators by generalized Stokes systems},
  author={Dietmar Gallistl},
  journal={Math. Comput.},
  • D. Gallistl
  • Published 29 March 2017
  • Mathematics, Computer Science
  • Math. Comput.
A stable splitting of 2m-th order elliptic partial differential equations into 2(m− 1) problems of Poisson type and one generalized Stokes problem is established for any space dimension d ≥ 2 and any integer m ≥ 1. This allows a numerical approximation with standard finite elements that are suited for the Poisson equation and the Stokes system, respectively. For some fourthand sixth-order problems in two and three space dimensions, precise finite element formulations along with a priori error… 

Figures and Tables from this paper

A framework to systematically decouple high order elliptic equations into combina4 tion of Poisson-type and Stokes-type equations is developed. The key is to systematically construct 5 the underling
Differential Complexes, Helmholtz Decompositions, and Decoupling of Mixed Methods
A framework to systematically construct differential complex and Helmholtz decompositions is developed, used to decouple the mixed formulation of high order elliptic equations into combination of Poisson-type and Stokes-type equations.
This paper studies formulations of second-order elliptic partial differential equations in nondivergence form on convex domains as equivalent variational problems and discovers a new symmetric formulation based on a least-squares functional.
Variational Formulation and Numerical Analysis of Linear Elliptic Equations in Nondivergence form with Cordes Coefficients
These formulations enable the use of standard finite element techniques for variational problems in subspaces of $H^2$ as well as mixed finite element methods from the context of fluid computations.
The conforming virtual element method for polyharmonic problems
Mixed Finite Element Approximation of the Hamilton--Jacobi--Bellman Equation with Cordes Coefficients | SIAM Journal on Numerical Analysis | Vol. 57, No. 2 | Society for Industrial and Applied Mathematics
A mixed finite element approximation of H2 solutions to the fully nonlinear Hamilton– Jacobi–Bellman equation, with coefficients that satisfy the Cordes condition, is proposed and analyzed. A priori
Design and convergence analysis of the conforming virtual element method for polyharmonic problems
The conforming virtual element method is developed and analyzed for the numerical approximation of polyharmonic boundary value problems, and an abstract result is proved that states the convergence of the method in the energy norm.
A Morley-Wang-Xu element method for a fourth order elliptic singular perturbation problem
A Morley-Wang-Xu (MWX) element method with a simply modified right hand side is proposed for a fourth order elliptic singular perturbation problem, in which the discrete bilinear form is standard as
Numerical approximation of planar oblique derivative problems in nondivergence form
A priori and a posteriori error estimates as well as numerical computations on uniform and adaptive meshes are provided for approximating a uniformly elliptic oblique derivative problem in two-dimensional simply-connected domains.
Ultraweak formulation of linear PDEs in nondivergence form and DPG approximation


Finite element methods for elliptic equations using nonconforming elements
A finite element method is developed for approximating the solution of the Dirichlet problem for the biharmonic operator, as a canonical example of a higher order elliptic boundary value problem. The
Minimal finite element spaces for 2m-th-order partial differential equations in Rn
The infinite element spaces constructed in this paper constitute the only class of finite element spaces that are known and proven to be convergent for the approximation of any 2m-th-order elliptic problems in any Rn, such that n ≥ m ≥ 1.
Finite element approximation of eigenvalue problems
  • D. Boffi
  • Mathematics, Computer Science
    Acta Numerica
  • 2010
The final part tries to introduce the reader to the fascinating setting of differential forms and homological techniques with the description of the Hodge–Laplace eigenvalue problem and its mixed equivalent formulations.
Finite element approximation of a sixth order nonlinear degenerate parabolic equation
In addition to showing well-posedness of the approximation, this work proves convergence in space dimensions of the sixth order nonlinear degenerate parabolic equation.
Morley Finite Element Method for the Eigenvalues of the Biharmonic Operator
New eigenvalue error estimates for nonconforming finite elements that bound the error of (possibly multiple or clustered) eigenvalues by the approximation error of the computed invariant subspace are presented.
Conforming and nonconforming finite element methods for solving the stationary Stokes equations I
Both conforming and nonconforming finite element methods are studied and various examples of simplicial éléments well suited for the numerical treatment of the incompressibility condition are given.
An interior penalty method for a sixth-order elliptic equation
A C(0) interior penalty method for a sixth-order elliptic equation on polygonal domains is derived and it is proved the well-posedness of the method as well as derive quasi-optimal error estimates in the energy norm.
C 0 Interior Penalty Galerkin Method for Biharmonic Eigenvalue Problems
We consider the C0 interior penalty Galerkin method for biharmonic eigenvalue problems with the boundary conditions of the clamped plate, the simply supported plate and the Cahn-Hilliard type. We
Mixed Finite Element Methods and Applications
Preface.- Variational Formulations and Finite Element Methods.- Function Spaces and Finite Element Approximations.- Algebraic Aspects of Saddle Point Problems.- Saddle Point Problems in Hilbert