A Priori Error Estimates of Finite Element Methods for Linear Parabolic Integro-Differential Optimal Control Problems

2014 ◽  
Vol 6 (5) ◽  
pp. 552-569 ◽  
Author(s):  
Wanfang Shen ◽  
Liang Ge ◽  
Danping Yang ◽  
Wenbin Liu
Keyword(s):  
Optimal Control ◽  
Finite Element ◽  
Error Estimates ◽  
A Priori ◽  
Mathematical Formulation ◽  
Finite Element Approximation ◽  
A Priori Error Estimates ◽  
Element Approximation ◽  
Weak Formulation ◽  
Priori Error Estimates

AbstractIn this paper, we study the mathematical formulation for an optimal control problem governed by a linear parabolic integro-differential equation and present the optimality conditions. We then set up its weak formulation and the finite element approximation scheme. Based on these we derive the a priori error estimates for its finite element approximation both in H1 and L2 norms. Furthermore some numerical tests are presented to verify the theoretical results.

scholarly journals A priori error estimates for the space-time finite element approximation of a quasilinear gradient enhanced damage model

2021 ◽  
Author(s):  
Marita Holtmannspötter
Keyword(s):  
Finite Element ◽  
Error Estimates ◽  
A Priori ◽  
Damage Model ◽  
Space Time ◽  
A Priori Error Estimates ◽  
Element Approximation ◽  
Element Discretization ◽  
Priori Error Estimates ◽  
Almost All

In this paper we investigate a priori error estimates for the space-time Galerkin finite element discretization of a quasilinear gradient enhanced damage model. The model equations are of a special structure as the state equation consists of two quasilinear elliptic PDEs which have to be fulfilled at almost all times coupled with a nonsmooth, semilinear ODE that has to hold true in almost all points in space. The system is discretized by a constant discontinuous Galerkin method in time and usual conforming linear finite elements in space. Numerical experiments are added to illustrate the proven rates of convergence.

A mixed finite-element method for elliptic operators with Wentzell boundary condition

2018 ◽  
Vol 40 (1) ◽  
pp. 87-108
Author(s):  
Eberhard Bänsch ◽  
Markus Gahn
Keyword(s):  
Boundary Condition ◽  
Finite Element ◽  
Mixed Finite Element ◽  
A Priori ◽  
Mixed Finite Element Method ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Weak Formulation ◽  
Wentzell Boundary Condition ◽  
Polyhedral Domain

Abstract In this paper we introduce and analyze a mixed finite-element approach for a coupled bulk-surface problem of second order with a Wentzell boundary condition. The problem is formulated on a domain with a curved smooth boundary. We introduce a mixed formulation that is equivalent to the usual weak formulation. Furthermore, optimal a priori error estimates between the exact solution and the finite-element approximation are derived. To this end, the curved domain is approximated by a polyhedral domain introducing an additional geometrical error that has to be bounded. A computational result confirms the theoretical findings.

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