An adaptive finite element method for laser surface hardening of steel problem

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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

A priori error estimates for the optimal control of laser surface hardening of steel

A priori error estimates for the optimal control of laser surface hardening of stee

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

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

Convergence of an adaptive mixed finite element method for general second order linear elliptic problems

The convergence of an adaptive mixed finite element method for general second order linear elliptic problems defined on simply connected bounded polygonal domains is analyzed in this paper. The main difficulties in the analysis are posed by the non-symmetric and indefinite form of the problem along with the lack of the orthogonality property in mixed finite element methods. The important tools in the analysis are a posteriori error estimators, quasi-orthogonality property and quasi-discrete reliability established using representation formula for the lowest-order Raviart-Thomas solution in terms of the Crouzeix-Raviart solution of the problem. An adaptive marking in each step for the local refinement is based on the edge residual and volume residual terms of the a posteriori estimator. Numerical experiments confirm the theoretical analysis.Comment: 24 pages, 8 figure

Dual weighted residual method for laser surface hardening of steel problem

Abstract. The main focus of this article is on the development of Adaptive Finite Element Method (AFEM) for the optimal control problem of laser surface hardening of steel governed by a dynamical system consisting of a semi-linear parabolic equation and an ordinary differential equation using Dual Weighted Residual Method (DWR). A posteriori error estimators using DWR method have been developed when a continuous piecewise linear discretization has been used for the finite element approximation of space variables and a discontinuous Galerkin method has been used for time and control discretizations. Further numerical results obtained are presented are compared with residual method numerical results. Key Words. Laser surface of steel problem, Adaptive finite element methods, Dual weighted residual methods, a posteriori error estimates. 1