Numerical Solution of Partial Differential Equations, Fundamentals of Finite Element Method

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Department of Technical Mathematics

Synopsis:

Mathematical background of the finite element method. Banach and Hilbert spaces. Linear forms, bilinear forms, scalar product. Hölder and Cauchy inequality. Lax-Milgram theorem. Lebesgue and Sobolev spaces. Sobolev imbeddings theorem and the trace theorem. Green theorem. Substitution theorem. Poincare-Friedrichs inequality.

Basic principle of the finite element method. Example of application for 1D problem, classical and weak solution, error estimates. Abstract variational formulation, Ritz and Galerkin problem. Existence and uniquness of the solution. Discrete Ritz and Galerkin problems. Cea's lemma (error estimate).

Application of finite element method for 2D problem. Weak formulation for the case of zero Dirichlet boundary condition. Discretization using Lagrangiang linear elements, finite element space and base construction. Assembling of the stiffness matrix and the load vector. Weak formulation for mixed boundary conditions. Reference element and mapping, extension to 3D and higher order finite elements.

Solution of the discrete problem – systems of linear equations. Direct and iterative methods. Gradient methods, conjugate gradient method and preconditioning.

Requirements:

Syllabus of lectures:

1 - 3. Principle properties of finite diference method in partial differential equations.

4 - 6. Variational formulation of boundary value problems for partial differential equations,

Weak solution, mathematical principles of the finite element Metod (FEM).

7 - 9. FEM for elliptic, parabolic and hyperbolic equations. Examples in 1D, 2D.

10 - 12. Algorithms in FEM. Examples for individual work.

13 - 14. FEM for nonlinear problems. Software for FEM. Show of applications of FEM.

Syllabus of tutorials:

1 - 3. Principle properties of finite diference method in partial differential equations.

4 - 6. Variational formulation of boundary value problems for partial differential equations,

Weak solution, mathematical principles of the finite element Metod (FEM).

7 - 9. FEM for elliptic, parabolic and hyperbolic equations. Examples in 1D, 2D.

10 - 12. Algorithms in FEM. Examples for individual work.

13 - 14. FEM for nonlinear problems. Software for FEM. Show of applications of FEM.

Study Objective:

Mathematical background of the finite element method. Banach and Hilbert spaces. Linear forms, bilinear forms, scalar product. Hölder and Cauchy inequality. Lax-Milgram theorem. Lebesgue and Sobolev spaces. Sobolev imbeddings theorem and the trace theorem. Green theorem. Substitution theorem. Poincare-Friedrichs inequality.

Basic principle of the finite element method. Example of application for 1D problem, classical and weak solution, error estimates. Abstract variational formulation, Ritz and Galerkin problem. Existence and uniquness of the solution. Discrete Ritz and Galerkin problems. Cea's lemma (error estimate).

Application of finite element method for 2D problem. Weak formulation for the case of zero Dirichlet boundary condition. Discretization using Lagrangiang linear elements, finite element space and base construction. Assembling of the stiffness matrix and the load vector. Weak formulation for mixed boundary conditions. Reference element and mapping, extension to 3D and higher order finite elements.

Solution of the discrete problem – systems of linear equations. Direct and iterative methods. Gradient methods, conjugate gradient method and preconditioning.

Study materials:

C.Johnson: Numerical Solution of Partial Differential Equation by the Finite Element Method, Cambridge University Press, 1987

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No time-table has been prepared for this course

The course is a part of the following study plans: