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Euler, Schwarz, Ritz, Galerkin Martin J. Gander Before Ritz From Euler, Schwarz, Ritz and Galerkin Brachystochrone Euler Lagrange to Modern Computing Laplace Riemann Schwarz Runge/Ritz Ritz Martin J. Gander Vaillant Prize Chladni Figures Ritz Method martin.gander@unige.ch Results Road to FEM Timoshenko University of Geneva Bubnov Galerkin Courant RADON Colloquium, November, 2011 Clough Concluding Example In collaboration with Gerhard Wanner

Euler, Schwarz, Brachystochrone Ritz, Galerkin ( βραχυς = short, χρ o ν o ς = time) Martin J. Gander Before Ritz Johann Bernoulli (1696) , challenge to his brother Jacob: Brachystochrone Euler “Datis in plano verticali duobus punctis A & B, Lagrange Laplace assignare Mobili M viam AMB, per quam gravitate Riemann Schwarz sua descendens, & moveri incipiens a puncto A, Runge/Ritz Ritz brevissimo tempore perveniat ad alterum punctum Vaillant Prize B.” Chladni Figures Ritz Method Results Road to FEM A x Timoshenko Bubnov Galerkin Courant Clough Concluding dx dx Example dy dy ds ds M B y See already Galilei (1638)

Euler, Schwarz, Mathematical Formulation Ritz, Galerkin Martin J. Gander Letter of de l’Hˆ opital to Joh. Bernoulli, June 15th, 1696: Before Ritz Brachystochrone Euler Ce probleme me paroist des plus curieux et des Lagrange Laplace plus jolis que l’on ait encore propos´ e et je serois Riemann Schwarz bien aise de m’y appliquer ; mais pour cela il seroit Runge/Ritz necessaire que vous me l’envoyassiez reduit ` a la Ritz Vaillant Prize math´ ematique pure, car le phisique m’embarasse Chladni Figures Ritz Method . . . Results Road to FEM Time for passing through a small arc length ds : dJ = ds v . Timoshenko Speed (Galilei): v = √ 2 gy Bubnov Galerkin Courant Need to find y ( x ) with y ( a ) = A , y ( b ) = B such that Clough Concluding Example � b � b dx 2 + dy 2 � � 1 + p 2 ( p = dy J ( y ) = √ 2 gy = √ 2 gy dx = min dx ) a a

Euler, Schwarz, Euler’s Treatment Ritz, Galerkin Euler (1744) : general variational problem Martin J. Gander Before Ritz � b ( p = dy Brachystochrone J ( y ) = Z ( x , y , p ) dx = min dx ) Euler Lagrange a Laplace Riemann Schwarz Runge/Ritz Theorem (Euler 1744) Ritz The optimal solution satisfies the differential equation Vaillant Prize Chladni Figures Ritz Method Results N − d N = ∂ Z P = ∂ Z Road to FEM dx P = 0 where ∂ y , Timoshenko ∂ p Bubnov Galerkin Courant Clough Proof. Concluding Example

Euler, Schwarz, Joseph Louis de Lagrange Ritz, Galerkin August 12th, 1755: Ludovico de la Grange Tournier (19 Martin J. Gander years old) writes to Vir amplissime atque celeberrime L. Euler Before Ritz September 6th, 1755: Euler replies to Vir praestantissime Brachystochrone Euler atque excellentissime Lagrange with an enthusiastic letter Lagrange Laplace Riemann Idea of Lagrange: suppose y ( x ) is solution, and add an Schwarz arbitrary variation εδ y ( x ). Then Runge/Ritz Ritz � b Vaillant Prize Chladni Figures J ( ε ) = Z ( x , y + εδ y , p + εδ p ) dx Ritz Method Results a Road to FEM must increase in all directions , i.e. its derivative with respect Timoshenko Bubnov to ǫ must be zero for ǫ = 0: Galerkin Courant � b Clough ∂ J ( ε ) | ε =0 = ( N · δ y + P · δ p ) dx = 0 . Concluding ∂ε Example a Since δ p is the derivative of δ y , we integrate by parts: � b ( N − d dx P ) · δ y · dx = 0 a

Euler, Schwarz, Central Highway of Variational Calculus Ritz, Galerkin Martin J. Gander Since δ y is arbitrary, we conclude from Before Ritz Brachystochrone � b Euler ( N − d Lagrange dx P ) · δ y · dx = 0 Laplace Riemann a Schwarz Runge/Ritz that for all x Ritz N − d Vaillant Prize dx P = 0 Chladni Figures Ritz Method Results Central Highway of Variational Calculus: Road to FEM Timoshenko 1. J ( y ) − → min Bubnov Galerkin Courant dJ ( y + ǫ v ) ! 2. | ε =0 = 0: weak form Clough d ε Concluding 3. Integration by parts, arbitrary variation: strong form Example Connects the Lagrangian of a mechanical system (difference of potential and kinetic energy) to the differential equations of its motion. This later led to Hamiltonian mechanics.

Euler, Schwarz, Gravitation of a Complicated Body Ritz, Galerkin Newton (Principia 1687): inverse square law for celestial Martin J. Gander 1 bodies, f proportional to Before Ritz r 2 Brachystochrone ( x , y , z ) r Euler Lagrange ( ξ, η, ζ ) f Laplace Riemann Schwarz Runge/Ritz Ritz Vaillant Prize With f = ( f 1 , f 2 , f 3 ) (see also Euler 1749) we get Chladni Figures Ritz Method f 1 ≈ x − ξ � Results ( x − ξ ) 2 + ( y − η ) 2 + ( z − ζ ) 2 r := , Road to FEM r 3 Timoshenko Bubnov Laplace (1785): What if the celestial body is not a point ? Galerkin Courant Clough Concluding Example ��� ρ ( ξ, η, ζ ) x − ξ f 1 = d ξ d η d ζ r 3

Euler, Schwarz, Potential Function Ritz, Galerkin Idea of Laplace: Introduce the potential function Martin J. Gander Before Ritz ��� ρ ( ξ, η, ζ )1 Brachystochrone u = r d ξ d η d ζ Euler Lagrange Laplace Riemann Taking a derivative with respect x , we obtain Schwarz Runge/Ritz Ritz � ∂ u � ∂ r = − x − ξ 1 ∂ x , ∂ u ∂ y , ∂ u Vaillant Prize = ⇒ f = − r 3 Chladni Figures ∂ x ∂ z Ritz Method Results since e.g. Road to FEM Timoshenko Bubnov ��� ρ ( ξ, η, ζ ) x − ξ Galerkin f 1 = Courant d ξ d η d ζ r 3 Clough Concluding Example r 3 = r 3 − 3( x − ξ ) 2 r x − ξ ∂ Differentiating once more, we obtain r 6 ∂ x ∆ u = ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 + ∂ 2 u ∂ z 2 = 0!

Euler, Schwarz, Laplace’s Equation Ritz, Galerkin Martin J. Gander Laplace’s equation was also discovered independantly in: Before Ritz Brachystochrone Euler Lagrange ◮ theory of stationary heat transfer (Fourier 1822); Laplace Riemann Schwarz ◮ theory of magnetism (Gauss and Weber in G¨ Runge/Ritz ottingen Ritz 1839); Vaillant Prize Chladni Figures Ritz Method ◮ theory of electric fields (W. Thomson, later Lord Kelvin Results Road to FEM 1847, Liouville 1847); Timoshenko Bubnov Galerkin ◮ conformal mappings (Gauss 1825); Courant Clough ◮ irrotational fluid motion in 2D (Helmholtz 1858) Concluding Example ◮ in complex analysis (Cauchy 1825, Riemann Thesis 1851);

Euler, Schwarz, Riemann Mapping Theorem Ritz, Galerkin Riemann (Thesis 1851 in G¨ ottingen): Martin J. Gander “Eine vollkommen in sich abgeschlossene mathematische Theorie, Before Ritz Brachystochrone welche . . . fortschreitet, ohne zu scheiden, ob es sich um die Euler Lagrange Schwerkraft, oder die Electricit¨ at, oder den Magnetismus, oder Laplace Riemann das Gleichgewicht der W¨ arme handelt.” Schwarz Runge/Ritz Theorem Ritz If f ( z ) = u ( x , y ) + iv ( x , y ) is holomorph in Ω , then Vaillant Prize Chladni Figures Ritz Method Results ∆ u = u xx + u yy = 0 ∆ v = v xx + v yy = 0 . and Road to FEM Timoshenko Bubnov Galerkin Proof: Courant Clough � f ( z + h ) − f ( z ) = ∂ f df ( z ) lim h → 0 Concluding h ∂ x = = ⇒ u x + iv x = − iu y + v y f ( z + ih ) − f ( z ) Example = − i ∂ f dz lim h → 0 ih ∂ y

Euler, Schwarz, Conformal maps Ritz, Galerkin Martin J. Gander The Jacobian of such a function satisfies Before Ritz � ∂ u � ∂ u � cos φ � � ∂ u ∂ u Brachystochrone � sin φ Euler ∂ x ∂ y ∂ x ∂ y = = || ( u x , u y ) || · Lagrange ∂ v ∂ v − ∂ u ∂ u − sin φ cos φ Laplace ∂ x ∂ y ∂ y ∂ x Riemann Schwarz Runge/Ritz Ritz Vaillant Prize Chladni Figures Ritz Method Results Road to FEM Timoshenko Bubnov Galerkin Courant Clough Concluding Example “ ... und ihre entsprechenden kleinsten Theile ¨ ahnlich sind;” (Thesis § 21)

Euler, Schwarz, Riemann Mapping Theorem Ritz, Galerkin “Zwei gegebene einfach zusammenh¨ angende Fl¨ achen k¨ onnen stets Martin J. Gander so aufeinander bezogen werden, dass jedem Punkte der einen ein Before Ritz Brachystochrone mit ihm stetig fortr¨ uckender Punkt entspricht...;” Euler Lagrange Laplace Riemann Schwarz Runge/Ritz Ritz Vaillant Prize Chladni Figures Ritz Method Results Road to FEM Timoshenko Bubnov Galerkin Courant Clough Concluding Example (drawing M. Gutknecht 18.12.1975)

Euler, Schwarz, Idea of Riemann’s Proof Ritz, Galerkin Martin J. Gander Before Ritz Brachystochrone Euler Lagrange Laplace Riemann Schwarz Runge/Ritz Ritz Vaillant Prize Chladni Figures Ritz Method Results Find f which maps Ω to the unit disk and z 0 to 0: set Road to FEM Timoshenko f ( z ) = ( z − z 0 ) e g ( z ) , g = u + iv = Bubnov ⇒ z 0 only zero Galerkin Courant Clough In order to arrive on the boundary of the disk Concluding Example | f ( z ) | = 1 , z ∈ ∂ Ω = ⇒ u ( z ) = − log | z − z 0 | , z ∈ ∂ Ω . Once harmonic u with this boundary condition is found, construct v with the Cauchy-Riemann equations. Question: Does such a u exist ???