FEM (Ch5)

hp-Finite Elements (Supp)

Basics (Ch5.0)

Finite element error analysis (Ch5.2)

A posteriori error estimates (Ch5.3)

finite element system assembling (Ch5.1)

Non-conforming FEM (Ch5.4)

Intro (Ch0)

Legendre Polynomials (Ch1)

Projection based interpolation (Ch4)

Orthogonal polynomials on triangles (Ch3)

Err estimate of the $L_2$ projection (Ch2)

finite element $ (T, V_T, \Psi_T) $ (Ciarlet's Def59)

bound interpolation error
basis for error estimate
main application of Bramble-Hilbert (Le63)

equivalent (P46)

$ h_T $diameter of element $ T $ (P50)

finite element complex $ \{ (T,V_T,\Psi_T) \} $ (P46)

finite element space $ V_{\mathcal{T}}$

Graded meshes around vertex singularities (P55)

idea: bound approx. error by interpolation error (P49)

Finite element error estimate (Th65)
$ \left\Vert u-u_h \right\Vert_{H^1} \preceq h\left\Vert u \right\Vert_{H^2} $

Error estimates in $ L_2 $ norm P(52)

Approximation of Dirichlet BC (P53)
if $ A $ is coercive & continuous Th65 holds

Zienkiewicz Zhu EE (P56)

residual EE (P57)

goal driven EE (P61)

Mesh refinement algorithms (Application) (P62)

error estimator (EE) $ \eta (u_h, f) $ (P55)

High order elements (P54)
faster convergence with smoother solutions $ u $

w/ Dirichlet BC

The non-conforming $ P^1 $ triangle (P66)

2nd Lemma of Strang (Le74)

1st Lemma of Strang (Le73)

Def via Rodrigues formula
$ P_n (x) := \frac{1}{2^nn!} \frac{d^n}{dx^n} (x^2-1)^n$

w/o Dirichlet BC

formula $ (n+1)P_{n+1}(x) $ (Le2)

Sturm-Liouville diff. eq. satisfied (Le3)

Motivation: optimal balance of mesh size $h $ & polynomial order $ p $ to limit better than $ \inf_{v_{hp}\in V_{hp}} \Vert u-v_{hp} \Vert_{H^1}\leq ch^{m-1}\Vert u \Vert_{H^m} $

goal: p-version err estimate $\leq c(\frac{h}{p})^{m-1}\Vert u \Vert_{H^m} $

(JP) Jacobi Polynomials (P5)

hp-version would lead to exp. conv. $ \leq ce^{-N^{\alpha}}$

$ \int\limits^{1}_{-1}P_n(x)\,P_m(x)\, \textrm{dx}=\frac{2}{2n+1}\delta_{n,m}$ (Le1)

polynomials dense in $L_2(-1,1) \Rightarrow u=\sum\limits_{n=0}^{\infty}a_nP_n$
with $ a_n $ generalized Fourier coeffs.

approx. err estimates of orth. polynomials (Ch4.0)

$ P_{L_2}^{\Pi_p} $ is $ L_2 $ projection to ${\Pi_p} $

1D-case (Ch4.1)

$ T $ bounded set

$ V_T $ function space on $ T $

orthogonal w.r.t $ (u',v')_{L_2,1-x^2}$ as implication of (Le3)

JP are orthogonal w.r.t. the weighted IP

Lagrange finite elements

...on triangles (Ch4.2)

Examples

global interpolation operator $ I_{\mathcal{T}}v_{\vert T} $ (P47)

reference finite element (P46)

$ \Psi_T $ lin. independent functionals on $ V_T $

projection err is bounded (Le4)

shape regular (for families of triangulations)

Dubiner basis = $ L_2(T) $ orthogonal basis for $ \Pi_p(T)$ by JPs(Le5)

equivalent $ \Rightarrow $ interpolation equivalent (Le60)

affine equivalent

Aubin-Nitsche (Th66) + Proof
$ \left\Vert u-u_h \right\Vert_{L_2} \preceq h^2\left\Vert u \right\Vert_{H^2} $

Examples: $ P^1, P^0$ with different node locations

Red Green Refinement

connectivity matrix $ C_T \in \mathbb{R}^{N \times N_T} $ (P47)

regularity (P47)

Equilibrated Residual Error Estimates (Supp)

dual problem

marked edge bisection

Hermite finite elements

interpolation equivalent

efficient $ \left\Vert u-u_h \right\Vert_V \geq C_2 \eta (u_h, f) $

usual error estimators $ \eta^2 (u_h, f) = \sum_{T \in \mathcal{T}} \eta^2_T (u_h, f) $

Clement quasi-interpolation operator (Def69)

no requirement for $ V_h \subset V $

$ \Vert u-u_h \Vert_h \preceq \inf_{v_h \in V_h} \Vert u-v_h \Vert_h + \sup_{w_h \in V_h} \frac{A_h(u,w_h)-f_h/w_h)}{\Vert w_h \Vert_h}$

Assume: coerc. & cont. of $ A_h(.,.) $

$ C^m $ smooth functions

quasi uniform triangulation/mesh (P51)

(regular) triangulation $ \mathcal{T}=\{ T_1,...,T_M \} $ (P46)

pseudo code

reliability (Th71) + Proof

2nd Lemma of Strang gives error estimate $ \left\Vert u-u_h \right\Vert \preceq h \Vert u \Vert_{H^2} $

reliable $ \left\Vert u-u_h \right\Vert_V \leq C_1 \eta (u_h, f) $

or approx. by Robin BC

local efficiency estimates $ \left\Vert u-u_h \right\Vert_{H^1(\omega_T)} \geq C_2 \eta_T (u_h, f) $

numerical integration

difficult: interpolation operator to an $ H^1 $ conforming FE space

BLF & LF replaced by mesh dependent forms

Applications (NS-eq, $L_2$ Matrix is diag.)

projection based interp.

finite element problem

$ I_p$err estimates for $ u \in H^1, p \in L_2, q \in \Pi_p $

usual spaces for $ T \subset \mathbb{R}^2$

p-version Galerkin Method for Dirichlet problem through operator $ I_p$

point evaluations functionals

$ \exists $ extension $ \widetilde{v}\in \Pi_p(T) $ (Le7)

nodal basis $ \{\varphi_T^i \}$ for $ V_T $, dual to $ \Psi_T$

linear or quadratic line segment

err estimate (which was the goal) (Le8)

Hermite line segment

computation of the lifting $ \Vert \sigma^{\Delta}\Vert $ (Ch2)

$ V_h^{nc}=\left\{ v \in L_2 : v_{\vert T} \in P^1 (T), v \mathrm{\,is\,cont.\, in\, edge\,midpoints} \right\}$

inserting results in reduced system

commuting diagram $ \Pi_{L_2}^{\Pi_{p-1}}u'=(I_pu)' $ (Le6)

triangle

local nodal interpolation operator $ I_T v$ (P46) (is projection)

interpolation equivalent

easy

obtain linear system

not equivalent

are equivalent

framework (Ch1)

using point eval. and derivatives

$ Q^p $

$ P^p $

explicit construction

Morely

constant

rewrite $ r() $ as functional

non-conforming

localize construction of flux

further notes/attrinbutes

linear

on n-dim simplices

Raviart-Thomas

applies also in 3D

residual $ r(\cdot)\in V^*$
$r(v)=f(v)-A(u_h,v),\,v\in V$

in 1D

goal: provide upper-bounds for discretization err w/o generic constants

on tensor product elements

main idea: calculate lifting $ \sigma^{\Delta}$

proves existence of equilibrated flux

of equilibrated fluxes on vertex patch

evaluation $L_2$ norm of $ \sigma^{\Delta}$ easy

such that $ \textrm{div}\sigma^{\Delta}=r$

of $ \sigma^{\Delta} $ in terms of BDM elements

instead: solve local optimization problem

res-err estimator w/ weighted $L_2$-norms

name: flux-postprocessing
$ \Rightarrow \textrm{div}\sigma=f \Rightarrow $ flux in exact equlibrium with source $ f $

different representation by sums of

problem: direct eval. of $ \Vert r \Vert$ not feasible

equlibrated res err estimator $ \eta^{er}$

edge-res (normal jump) $ r_E $

element-res $ r_T $

efficient w/ generic constant c

reliable w/ c=1