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| Assignments and schedule |
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- 1/7: Introduction and overview. L^p,C^k spaces. Read Chapter I from text.
- 1/9: Weak derivatives. H^m, W^{m,p} spaces. Model problem in 1D (D). Variational formulation (V).
- 1/11: Minimization problem (M). Equivalence of (V) and (M). Definition of (FE) solution
(Galerkin and Ritz).
- 1/14: (FE) solution (discrete system) versus FD solution.
- HW 1 due 1/28. You can use the code FEM1D.m
for a template of MATLAB solution.
- 1/16: Basic error estimate for FE solution in 1D. Interpolation in 1D. Poincare-Friedrichs constant.
- 1/18: Other problems in 1D, different boundary conditions. Numerical integration and how to use it with FEM1D.
- 1/23: Start Chapter II from book: boundary value problems in 1D-2D-3D. Meaning/desired properties of
the coefficient $k$ in $-\nabla \cdot ( k \nabla u)=f$. Bilinear form (coercivity and continuity).
- 1/25 Examples of forms and BVP. Lax-Milgram Theorem. (Chapter II.$1-3).
- 1/28-30 Construction of V_h in d=1,d=2: grids and meshes and piecewise polynomials (Chapter II.$4).
- 2/4: CLASS CANCELLED due to illness.
- 2/6: Affine families of finite elements. Estimates for shape-regular meshes. Integration on
reference element.
- 2/8: calculations on the reference element and using affine map. Local and global stiffness matrix, assembly
process.
- 2/11-13: general review of interpolaiton/approximation using polynomials
- HW 2 due 2/29. You should use ACF code; and you can also use
tri_quadcofs.m
for quadrature set-up on triangles. For rectangles, use tensor product quadrature in 1D as in FEM1D.m
- 2/15: Consequences of Bramble-Hilbert Lemma. Thm. 6.4 and 6.8
(interpolation error estimate and inverse estimate). Deriving error
for Galerkin solution to symmetric variational problems (revisited
ideas from 1D): (i) Galerkin orthogonality, (ii) stability estimate
and Cea's lemma, (iii) energy error estimate. Start L_2 error estimate
(Aubin-Nitsche = duality method).
- 2/18: finish Aubin-Nitsche and L_2 error estimates. Summary of results
and code construction for model problem.
- 2/20: non-homogenous Dirichlet b.cond.
- 2/22-25: handle problems with zero- and first-order etrms, with various b.c. Dependence
of estimates on the Peclet number. (Recall II.1-3)
- 2/25: Variational crimes 1-4. Read III.1
- 2/27: Variational crimes cd. L_2 and elliptic (Ritz) projection.
- 2/29: Time dependent (parabolic) problem: semi-discrete and fully discrete
solution. Rit projection and estimates.
- 3/3-5: Error estimates for time dpeendent problems.
- HW 3 due at the latest by the time/date of the Final
which is Thursday by 14:00 (no extensions).
- 3/5-7: A-posteriori error estimation. Read on Clement
interpolation operator: II.$6.
- 3/10-12: Finite Elements for Linear Elasticity (Son-Young Yi)
- 3/14: Wrap-up and review, extensions of models to nonlinear
problems. What are FE good for, and what they are not good
for. Extensions of standard conforming method (preview).
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