Седмичен изглед

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  • 1 октомври - 7 октомври

    Lecture 1. 

    • Piecewise linear functions. Interpolation. Nodal basis.

  • 8 октомври - 14 октомври

    Lecture 2.

    • A priori error estimates for piecewise linear interpolation. 
    • L2 projection

  • 15 октомври - 21 октомври

    Lecture 3. 

    • A priori error estimate for the L2 projection in V_h. 
    • Idea of FEM. 
    • FEM for 1D problems with homogeneous Dirichlet boundary conditions---algorithm and a priori error estimates in energy norm.

  • 22 октомври - 28 октомври

    Lecture 4. 

    • FEM in 1D with more general boundary conditions. 
    • A priori error estimates in H1 norm. Coercivity and continuity of the bilinear form.

  • 29 октомври - 4 ноември

    Lecture 5.

    • A priori error estimates in L2 norm. Nitsche's trick.
    • Triangulation in 2D.
    • Piecewise linear polynomials in two variables---interpolation and L2-projection. A priori error estimates.

  • 5 ноември - 11 ноември

    Lecture 6.

    • Preliminaries from vector calculus
    • FEM for 2D stationary problems with homogeneous Dirichlet boundary conditions---variational form, Ritz--Galerkin problem,obtaining the linear algebraic system; a priori error estimates in H1 norm.
    • FEM for 2D stationary problems with more general boundary conditions.

  • 12 ноември - 18 ноември

    Lecture 7

    • FEM for stationary 2D problems with more general boundary conditions (continued)
    • Element-wise computations in 1D
    • Element-wise computations in 2D---preliminaries from multivariable calculus.

  • 19 ноември - 25 ноември

    Lecture 8.

    • Elementwise computations in 2D. Change of variables to the standard triangular element.
    • Computing element mass and stiffness matrices. 
    • Shape functions for linear triangular elements.
    • General vector-matrix representation of the element mass and stiffness matrices.

  • 26 ноември - 2 декември

    Lecture 9.

    • Elementwise computation for more general boundary conditions. Local matrices, corresponding to the contribution from the boundary conditions (from integration by parts)
    • Quadrature formulas for the standard triangular element

  • 3 декември - 9 декември

    Lecture 10.

    • Assembling the linear system (Continued)---imposing Dirichlet boundary conditions
    • FEM for non-stationary problems---algorithmic foundations. Examples: 1D diffusion equation, 2D wave equation

  • 10 декември - 16 декември

    Lecture 11.

    • Stability and convergence of FEM for non-stationary problems.
    • General theory of FEM for elliptic problems--introduction. Abstract formulation of the variational problem.
    • Existence and uniqueness of the solution---Riesz representation theorem, Lax--Miglram theorem. Examples.

  • 17 декември - 23 декември

    Lecture 12.

    • A priori error estimates for the FEM solution of the abstract variational problem. Cea's lemma.
    • Bramble--Hilbert lemma.
    • A priori error estimates for the interpolant in P_k over the standard element.
    • A priori error estimate for the piecewise-polynomial interpolant of k-th degree

  • 7 януари - 13 януари

    Lecture 13.

    • Study of FEM for the general elliptic problem.
    • Nitsche's trick

  • 14 януари - 20 януари

    Lecture 14.

    • Chosen topics from the theory of Sobolev and Hilbert spaces.
    • FEM and boundary conditions---principal and natural boundary conditions.