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Solving Inverse PDE by the Finite Element Method
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http://hdl.handle.net/2077/30180

Title:  Solving Inverse PDE by the Finite Element Method 
Authors:  Amad, Yahya 
Issue Date:  3Sep2012 
Degree:  Student essay 
Abstract:  In this Master thesis project solving inverse PDE by the finite element method. An optimal control
problems subjected to PDE constraint with boundary conditions is given. Construct the variational form
then construct Lagrangian, which defined over whole space. Lagrangian function is function of three
variables which is defined on whole space, evaluate their partial derivatives, these set of equations are
the stationary point equations. Solve these stationary point equations individual, combine and into one
equation by finite element method. The error, convergence rate, objective function value, error of objective
function is also computed.
To calculate finite element solution used FEniCS with Python. Construct the programs in Python and
run in FEniCS. Following things are discussed in the following chapters:
Chapter 1. Discussion of optimal contol problem with PDE’s constraints.
Chapter 2. Discussion of the variational formulation and Lagrangian function.
Chapter 3. In this chapter discussion how to solve equations for stationary point by finite element method.
To solve equations for stationary point used Python computer program language with FEniCS. FEniCS
can be programmed in Python. FEniCS solves partial differential equations, this project used FEniCS to
finite element equations by finite element method.
In this project a quadratic objective function subjected to linear elliptical partial differential equation
with Neumann boundary condition is known, construct the variational form, Lagrangian function which is
defined over whole space, taken partial derivatives of this Lagrangian function which gives set of equations
are called stationary point equations, write stationary point equations as finite element solution then these
equations are called finite element equations. The stationary point equations are used to find the exact
solution whereas finite element equations are used to calculate finite element solution. Finally solve these
finite element equations as one equation by finite element method, use programming tool FEniCS with
Python.
The error analysis, convergence rate, objective function value and error of objective function are also
computed. The matrix form of stationary point equations are also calculated and show that it is indefinite
of saddle point form.... more 
URI:  http://hdl.handle.net/2077/30180 
Appears in Collections:  Masteruppsatser

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