10 research outputs found
Numerical analysis for the pure Neumann control problem using the gradient discretisation method
The article discusses the gradient discretisation method (GDM) for
distributed optimal control problems governed by diffusion equation with pure
Neumann boundary condition. Using the GDM framework enables to develop an
analysis that directly applies to a wide range of numerical schemes, from
conforming and non-conforming finite elements, to mixed finite elements, to
finite volumes and mimetic finite differences methods. Optimal order error
estimates for state, adjoint and control variables for low order schemes are
derived under standard regularity assumptions. A novel projection relation
between the optimal control and the adjoint variable allows the proof of a
super-convergence result for post-processed control. Numerical experiments
performed using a modified active set strategy algorithm for conforming,
nonconforming and mimetic finite difference methods confirm the theoretical
rates of convergence
Morley Type Virtual Element Method for Von K\'{a}rm\'{a}n Equations
This paper analyses the nonconforming Morley type virtual element method to
approximate a regular solution to the von K\'{a}rm\'{a}n equations that
describes bending of very thin elastic plates. Local existence and uniqueness
of a discrete solution to the non-linear problem is discussed. A priori error
estimate in the energy norm is established under minimal regularity assumptions
on the exact solution. Error estimates in piecewise and norm are
also derived. A working procedure to find an approximation for the discrete
solution using Newtons method is discussed. Numerical results that justify
theoretical estimates are presented.Comment: 23 pages, 6 figures, 6 tables Submitted to a journa
A posteriori error analysis for a distributed optimal control problem governed by the von Kármán equations
This article discusses the numerical analysis of the distributed optimal control problem governed by the von Karman equations defined on a polygonal domain in R 2 . The state and adjoint variables are discretised using the nonconforming Morley finite element method and the control is discretized using piecewise constant functions. A priori and a posteriori error estimates are derived for the state, adjoint and control variables. The a posteriori error estimates are shown to be efficient. Numerical results that confirm the theoretical estimates are presented
Unified a priori analysis of four second-order FEM for fourth-order quadratic semilinear problems
A unified framework for fourth-order semilinear problems with trilinear
nonlinearity and general source allows for quasi-best approximation with
lowest-order finite element methods. This paper establishes the stability and a
priori error control in the piecewise energy and weaker Sobolev norms under
minimal hypotheses. Applications include the stream function vorticity
formulation of the incompressible 2D Navier-Stokes equations and the von
K\'{a}rm\'{a}n equations with Morley, discontinuous Galerkin, interior
penalty, and weakly over-penalized symmetric interior penalty schemes. The
proposed new discretizations consider quasi-optimal smoothers for the source
term and smoother-type modifications inside the nonlinear terms.Comment: 31 page
Morley finite element method for the von Kármán obstacle problem
This paper focusses on the von Kármán equations for the moderately large deformation of a very thin plate with the convex obstacle constraint leading to a coupled system of semilinear fourth-order obstacle problem and motivates its nonconforming Morley finite element approximation. The first part establishes the well-posedness of the von Kármán obstacle problem and also discusses the uniqueness of the solution under an a priori and an a posteriori smallness condition on the data. The second part of the article discusses the regularity result of Frehse from 1971 and combines it with the regularity of the solution on a polygonal domain. The third part of the article shows an a priori error estimate for optimal convergence rates for the Morley finite element approximation to the von Kármán obstacle problem for small data. The article concludes with numerical results that illustrates the requirement of smallness assumption on the data for optimal convergence rate