Lecture 18 Finite Element Methods (FEM): Functional Spaces and Splines. Songting Luo. Department of Mathematics Iowa State University

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1 Lecture 18 Finite Element Methods (FEM): Functional Spaces and Splines Songting Luo Department of Mathematics Iowa State University MATH 481 Numerical Methods for Differential Equations Draft Songting Luo ( Department of Mathematics Iowa State University[0.5in] MATH481 MATH 481 Numerical Metho 1 / 11

2 Outline 1 Functional Spaces and Operators Songting Luo ( Department of Mathematics Iowa State University[0.5in] MATH481 MATH 481 Numerical Metho 2 / 11 Draft

3 Function Spaces and Operators For simplicity, we assume the functions are defined on r0, 1s (extension to any interval ra, bs is straightforward). Function spaces and operators A function space is a vector space (linear space) whose elements are functions. For example: C k r0, 1s tk times continuously differentiable functions on r0, 1su C 0 r0, 1s tcontinuous functions on r0, 1su Most function spaces are infinite-dimensional. Draft Songting Luo ( Department of Mathematics Iowa State University[0.5in] MATH481 MATH 481 Numerical Metho 3 / 11

4 Function Spaces and Operators cont ed Inner Products and Norms Examples of Standard Inner Products: ă u, v ą 0 ş 1 0 upxqvpxqdx ă u, v ą 1 ş 1 0 ru1 pxqv 1 pxq ` upxqvpxqsdx ă u, v ą 2 ş 1 0 ru2 pxqv 2 pxq ` u 1 pxqv 1 pxq ` upxqvpxqsdx etc. It is not the previous p-norm! The subscript refers to how many derivatives are used! Each inner product generates a corresponding norm: }u} p? ă u, u ą p Draft Songting Luo ( Department of Mathematics Iowa State University[0.5in] MATH481 MATH 481 Numerical Metho 4 / 11

5 Function Spaces and Operators cont ed Operator An operator L between two function spaces is linear if Lpαu ` βvq αlpuq ` βlpvq The adjoint operator L is defined by ă Lu, v ą ă u, L v ą L is self-adjoint if L L. L is positive definite if ă Lu, u ąě and ă Lu, u ą 0 if and only if u 0. Draft Songting Luo ( Department of Mathematics Iowa State University[0.5in] MATH481 MATH 481 Numerical Metho 5 / 11

6 Function Spaces and Operators cont ed Examples For a finite-dimensional vector space, L is a matrix, L L T p transpose q L is positive definite ô all eigenvalues of L are positive. Consider Lupxq u 2 pxq, i.e., L : C k Ñ C k 2 is linear. It is self-adjoint on this smaller space Then ă Lu, v ą C k 0 r0, 1s tu P C k r0, 1s : up0q up1q 0u. ż 1 0 u 2 pxqvpxqdx ż 1 Draft 0 upxqv 2 pxqdx ă u, Lv ą, which is obtained by integration by parts. Songting Luo ( Department of Mathematics Iowa State University[0.5in] MATH481 MATH 481 Numerical Metho 6 / 11

7 Function Spaces and Operators cont ed Examples cont ed This L is also positive definite on C k 0 : ă Lu, u ą ż 1 0 u 1 pxq 2 dx using Integration by parts ă Lu, u ą 0 ñ u 1 0 ñ u 0. If enforcing up0q 0, u 1 p1q 0 or u 1 p0q 0, up1q 0 instead, similar conclusions hold. If enforcing u 1 p0q u 1 p1q 0, L is self-adjoint, but not positive definite. Draft Songting Luo ( Department of Mathematics Iowa State University[0.5in] MATH481 MATH 481 Numerical Metho 7 / 11

8 Function Spaces and Operators cont ed Sobolev Spaces A sequence tx n u P R converges to ą 0, DN so that x n M ă ɛ if n ą N. A sequence tx n u is a Cauchy ą 0, DN so that x n x m ă ɛ if n, m ą N. A normed vector space is complete if every Cauchy sequence converges. For examples: Q t rational numbers u is not complete. R, C, R n are complete. For an infinite-dimensional space, it could be complete in one norm, but not in another. For example, C 0 is complete in the 8-norm: }u} 8 max xpr0,1s t upxq u, but not in 0-norm: }u} 0 Draft b ş1 0 upxq 2 dx. Songting Luo ( Department of Mathematics Iowa State University[0.5in] MATH481 MATH 481 Numerical Metho 8 / 11

9 Function Spaces and Operators cont ed Sobolev Spaces The Sobolev space H 0 is the completion of C 0 in }u} 0 norm. It is also called L 2 (square integrable functions) The Sobolev space H k is the completion of C k in }u} k norm. The functions in H k are k times differentiable in some sense, but not in the classical sense Convergence in H k means that the function and all derivatives up to order k converge in the L 2 -sense. We want to work in H k instead of C k is due to the completeness. For a sequence of approximate solutions, we want the limit to exist in the same space. Draft Songting Luo ( Department of Mathematics Iowa State University[0.5in] MATH481 MATH 481 Numerical Metho 9 / 11

10 Splines, Piecewise Polynomial Interpolation Given points (nodes) x 0 ă x 1 ă ă x n, equally spaced or not. Splines S p k t piecewise polynomial of degree p with k continuous derivativeu For examples: S0 1 : piecewise constant polynomial, discontinuous. What are the basis functions? S0 1 : piecewise linear polynomial, continuous. What are the basis functions? S2 3 : piecewise cubic polynomial, two continuous derivatives. What are the basis functions? Draft Songting Luo ( Department of Mathematics Iowa State University[0.5in] MATH481 MATH 481 Numerical Metho 10 / 11

11 Splines cont ed Accuracy If tx i u are equally spaced with step-size h, then error in spline approximation from S p k is Ophp`1 q Splines To Use S0 1 : good for integral equations, but not differential equations. They are only H 0, not H 1 S 1 0 : it is P H1. (We will use it) S 3 2 : it is P H3. (work well in 1-D, more complicated though.) Draft Songting Luo ( Department of Mathematics Iowa State University[0.5in] MATH481 MATH 481 Numerical Metho 11 / 11

12 Overview: PDEs in Applications

13 Ex: Poisson s equation This equation smooths the solution: the solution times more differentiable than the source term is two Modeling: gravitation, ground water flow, electrostatics,... Johan Hoffman KTH p.7

14 Johan Hoffman KTH p.8 Dirichlet BC for and for BC:

15 Dirichlet + Neumann BC BC: for and for Johan Hoffman KTH p.9

16 Ex: Heat equation This equation smooth the solution over time: Modeling: heat conduction, pollution,... Johan Hoffman KTH p.10

17 Ex: Heat equation Johan Hoffman KTH p.11

18 Ex: Heat equation Osmosis; diffusion through cell membrane Johan Hoffman KTH p.12

19 Ex: Schrödinger equation is the wave function in the quantum mechanical model of the motion of an electron orbiting around one proton at the origin. Johan Hoffman KTH p.13

20 Ex: Black-Scholes equation This equation is close to the heat equation, with the asset price playing the role of the spatial dimension Modeling: option pricing,... Johan Hoffman KTH p.14

21 Ex: Linear Elasticity Cauchy-Navier s elasticity equations: div div Models the displacement elastic bodies and the stress for Johan Hoffman KTH p.15

22 Ex: Wave equation This equation conserves energy (for ): Modeling: wave phenomena, accoustics,... Johan Hoffman KTH p.16

23 Ex: Wave equation Seismic waves in simulation of California Earthquake Johan Hoffman KTH p.17

24 Ex: Transport equation The solution convection field is transported (convected) by the. Modeling: Pollution,... Johan Hoffman KTH p.18

25 Ex: Convection-Diffusion-Reaction This equation is a combination of transport (convection) by the convection field and reaction with reaction coefficient., diffusion with diffusivity Modeling: chemical reactions, pollution,..., of a spieces, Johan Hoffman KTH p.19

26 Ex: Convection-Diffusion-Reaction Chernobyl 1986; simulation by SMHI Johan Hoffman KTH p.20

27 Johan Hoffman KTH p.21 Ex: Maxwell equations current density electric field, magnetic field,

28 Ex: Maxwell equations Magnetic field around a coil. Johan Hoffman KTH p.22

29 Ex: Stokes equations velocity and pressure Modeling: low velocity flow phenomena Johan Hoffman KTH p.23

30 Ex: Stokes equations Groundwater flow Johan Hoffman KTH p.24

31 Ex: Navier-Stokes equations velocity and pressure Modeling: flow phenomena, weather prediction, blood flow,... Johan Hoffman KTH p.25

32 Ex: Navier-Stokes equations Johan Hoffman KTH p.26

33 Ex: Navier-Stokes equations Vorticity around wheel. Johan Hoffman KTH p.27

34 Ex: Navier-Stokes equations Vorticity around a full car. Johan Hoffman KTH p.28

35 Ex: Navier-Stokes equations Vorticity around a full car. Johan Hoffman KTH p.29

36 Ex: Navier-Stokes equations Blood flow in artery. Johan Hoffman KTH p.30