Inside Finite Elements
Summer term 2011, TU BerlinM. Weiser
News
- A reference picture for the DLY error distribution of task 4 in homework 6 is available below.
Dates and Rooms
| What | When | Where | Who |
|---|---|---|---|
| Lecture | We 10-12 Fr 16:00-17:30 | MA 648 MA 648 | M. Weiser |
| Exercise | We 16:00-17:30 | MA 541 | M. Weiser |
| Office hour | Tu 14-16 | MA 676 | M. Weiser |
| Secretariat | MA 674 | P. Grimberger |
Contents
All relevant implementation aspects of finite element methods are discussed in this course. The focus is on algorithms and data structures as well as on their concrete implementation. Theory is only covered as far as it gives insight into the construction of algorithms. In the homework, a complete FE-solver for scalar 2D problems will be implemented in Matlab/Octave.
The contents of lectures and homework will be coordinated with the course Numerical analysis of partial differential equations (K. Schmidt), such that attending both courses makes sense.
The course will be given in English.
Literature
-
Numerische Mathematik 3. Adaptive Lösung partieller Differentialgleichungen. de Gruyter. 2011.
- C. Grossmann, H.-G. Roos: Numerische Behandlung partieller Differentialgleichungen
- D. Braess: Finite Elemente
- J.-L. Guermond, A. Ern: Theory and Practice of Finite Elements
In-depth treatment, in particular part III on FE realization is relevant - J. Fish, T. Belytschko: A First Course in Finite Elements
Introductory text from an engineering point of view, almost no realization of FE. - C. Johnson: Numerical Solution of Partial Differential Equations by the Finite Element Method
Classic introductory textbook. - H.R. Schwarz: Finite Element Methods
Classic textbook.
Homework
- Exercise due to April 25
- Exercise due to May 2
- Exercise due to May 16
- Exercise due to May 23
Here is the boundary mass matrix for some grid with elements and nodes. - Exercise due to May 30
- Exercise due to June 13
Here are nodes and weights for a 6-point quadrature rule of order 4 on the reference triangle (from here). Here are the Gauss quadrature nodes and weights for quadrature over the reference line.
If your pictures for task 4 look like this one, your DLY implementation is probably correct. - Exercise due to June 20
- Exercise due to June 27
Here is a simple Jacobi smoother - Exercise due to July 4
Reference Implementation
- readPolyMesh.m
- plotMesh.m -- simple mesh visualization
- writeVTK.m
- numberOfElements.m
- numberOfVertices.m
- numberOfBoundaryElements.m
- transformationMatrix.m
- massMatrix.m
- stiffnessMatrix.m
- computeBoundary.m
- boundaryMassMatrix.m
- cg.m -- conjugate gradient method
- dly.m -- hierarchical error estimator
- jac.m -- Jacobi smoother
- longestEdge.m -- finds the longest edges of a triangle
- leppTerminal.m -- finds the end of the longest edge propagation path
- refine.m -- mesh refinement by longest edge bisection
- prolongation.m -- prolongation between levels by linear interpolation
- mgStack.m -- prolongations and projected Galerkin matrices for multilevel mesh hierarchies
- mgm.m -- classical V-cycle
Syllabus
The following schedule of topics is preliminary and subject to change.| When | What |
|---|---|
| We 13.4. | Basic equations prototypes, boundary conditions, classification |
| Fr 15.4. | Classical results, variational formulation of elliptic equations minimization, boundary conditions, Weierstrass, Lax-Milgram |
| We 20.4. | Ritz-Galerkin method orthogonality, bestapproximation, 1D simple finite elements, element matrices for piecewise constant coefficients |
| Fr 22.4. | --- |
| We 27.4. | |
| Fr 29.4. | |
| We 4.5. | 2D Grids and simple finite elements piecewise linear FE, conforming grids, data structures, element matrices for piecewise constant coefficients |
| Fr 6.5. | Sparse matrix data structures, performance of memory bandwidth/latency 3D grids, hanging nodes, nonsimplicial grids, grid notation |
| We 11.5. | a priori error estimates Solution of linear equations, direct methods |
| Fr 13.5. | --- |
| We 18.5. | classical iterative methods, gradient method |
| Fr 20.5. | preconditioning, CG method, termination |
| We 25.5. | Polynomial FE on simplices choice of basis, Lagrange, hierarchical, management of degrees of freedom, assembly Quadrature on quadrilaterals and simplices |
| Fr 27.5. | A posteriori error estimators residual, hierarchical |
| We 1.6. | Error estimators DLY, algorithmic realization, gradient recovery |
| Fr 3.6. | --- |
| We 8.6. | Grid refinement marking strategies, bisection, red-green |
| Fr 10.6. | Grid refinement LEPP backwards bisection refinement algorithm, grid hierarchies |
| We 15.6. | Classical Multigrid smoothing property, prolongation/restriction |
| Fr 17.6. | Classical Multigrid recursion, convergence |
| We 22.6. | Subspace correction methods abstract, domain decomposition, FE of higher order |
| Fr 24.6. | Subspace correction methods FE of higher order, HB |
| We 29.6. | Subspace correction methods: BPX Cascadic multigrid |
| Fr 1.7. | Newton's method for elliptic problems affine invariance, inexactness, globalization |
| We 6.7. | Parabolic problems abstract initial value problems, linear-implicit integrators |
| Fr 8.7. | Parabolic problems method of lines and method of time layers |
| We 13.7. | Case studies myocard excitation |
| Fr 15.7. |
Exam
- admittance: 50% of homework
- oral exam