Numerical Solution of partial differential equations-learning Summary

Tag: III indicates that play continuously produces the time class amp in the middle order

Partial Differential Equation numerical solution --- learning Summary 1. Knowledge Review (Note: \ (\ mit V \) is a linear space)

• Inner Product $ (\ cdot, \ cdot): \ mit V \ times \ mit V \ longrightarrow \ mathbb {r} $ is a bilinear ing and satisfies \ (I) (u, v) = (V, u), \ forall \, U, V \ In \ mit V \);

  $ (II) (u, u) \ GE 0, \ forall, U \ In \ mit v $; \ (III) (u, u) = 0 \) only when \ (u = 0 \).

• Semi-norm \ (| \ cdot |: \ mit V \ longrightarrow \ mathbb {r} \) is a linear ing that satisfies \ (I) | v | \ GE 0, \ forall \, V \ In \ mit V; \) $ (II) | CV | = | c | v |, \ forall V \ In \ mit V, \ forall C \ In \ mathbb {r}; $

  ? \ (Iii) | u + v | \ Leq | u | + | v |, \ forall U, V \ In \ mit v .\)

• Norm half-norm + condition: \ (| u | = 0 \) When and only when \ (u = 0 \).

• Norm Equivalence Theorem: Set \ (| \ cdot | \) and \ (| \ cdot | \) to linear space \ (\ mathbf V \) if there are two positive numbers \ (C_1 \) and \ (C_2 ,\)

  ? If the following unequal expressions are met, \ (| \ cdot | \) and \ (| \ cdot | \) are equivalent,
  \ [C_1 | v | \ Leq C_2 | v |, \ forall \, C_1, c_2 \ In \ mathbb {r}, \, \ forall \, V \ In \ mit v. \]

• Inner Product space if a linear space is given inner product, it is an inner product space.

  • Inner Product can generate induction norms
    \ [| V | = (V, V) ^ {1/2}, \, \ forall \, V \ In \ mit v. \]

  • Schwarz Inequality
    \ [| (W, V) | \ Leq | w | \, | v |, \, \ forall W, V \ In \ mit v. \]

• The complete inner product space of the Hilbert space is the Hilbert space, that is, any kernel sequence in the inner product space is converged.
• Norm linear space if a linear space is assigned a norm, it becomes norm linear space.
  • The inner product space must be a norm linear space, and its norm is \ (| v | = (V, V) ^ {1/2}, \, \ forall \, V \ In \ mit v. \)
• The complete norm linear space in the he space, that is, any sequence of the keys in the norm linear space is converged.
  • The Hilbert space must be the space of the East Coast.

2. New Concept

• Dual Space

  • If (\ mit V, | \ cdot | _ {\ mit v} \) and (\ mit W, | \ cdot | _ {\ mit w} \) is two norm linear spaces, from \ (\ mathbf V \) to \ (\ mit w \) the linear function of constitutes a norm linear space, which is recorded as \ (\ scr l (\ mit V; \ mit W )\). for \ (L \ In \ scr l (\ mit V; \ mit W) \), the defined norms are as follows:
    \ [| L | _ {\ scr l (\ mit V; \ mit W )}: = \ sup _ {0 \ neq v \ In \ mit v} \ frac {| LV |||_{\ mit W }{| | V ||_{\ MIT V }}. \]

  • If the \ (\ mit w \) space is a private space, \ (\ scr l (\ mit V; \ mit W) \) is also a private space.

  • If \ (\ mit W = \ mathbb {r} \), \ ({\ color {red} \ scr l (\ mit V; \ mathbb {r })} \) is the dual space of \ (\ mit V \). It is often recorded as \ ({\ color {red} \ mit V '}\).

  • Dual pair (Duality pairingIs called the dual pair between \ (\ mit V \) and \ (\ mit V,
    \ [\ Begin {Align *} <\ cdot \, \, \ cdot> &: \ mit V' \ times \ mit V \ longrightarrow \ mathbb {r }\& <L, V> \ longmapsto L (V ). \ end {Align *} \]

• Various convergence Definitions

  • Strong Convergence: The sequence \ (\ {v_n \} \) in the linear space \ (\ mit V \) is weak in the \ (V \, \ In \ mit V \) refers to the convergence by the norm, namely \ (| v_n-v | \ rightarrow0 (n \ rightarrow \ infty ). \)

  • Weak Convergence: The sequence \ (\ {v_n \} \) in the linear space \ (\ mit V \) is weak in the \ (V \, \ In \ mit V \) it refers to the convergence of \ (L (v_n) \) to \ (L (V), \) for any \ (L \ In \ mit V ),\) that is, \ (| L (v_n)-L (v) | \ rightarrow0 (n \ rightarrow \ infty ). \)

  • * Weak Convergence: The sequence \ (\ {l_n \} \) in the dual space \ (\ mit V' \) is weak in \ (L \, \ In \ mit V' \) refers to any one \ (V \ In \ mit V \), all

    ? \ (| L_n (V)-L (v) | \ rightarrow 0 (n \ rightarrow \ infty ).\)

    • \ (Strong convergence of {\ color {red} \ mit V \ rightarrow weak convergence .}\)

    • \ (Weak convergence in {\ color {red} \ mit V' \ rightarrow * Weak Convergence .}\)

• \ (\ Mit L ^ {p} (\ Omega )? \) Space \ (\ Omega _ {open} \ subset \ mathbb {r} ^ {d }? \), \ (D \ Ge 1? \), And \ (\ Omega? \) YesLebesgueMeasurable.
  \ [\ Begin {Align *} \ mit L ^ {p} &: = \ left \ {v \, \, \ big | \ int _ {\ Omega }\, \ left | V (x) \ right | \, ^ p \, DX \ Leq \ infty \ right \}, \, 1 \ Leq p \, <\ infty, \ mit L ^ {\ infty} &: = \ sup \ left \ {| V (x) | \ big | \,\, X \ In \ Omega \ right \} <\ infty. \ end {Align *} \]

  Its norm is
  \ [\ Begin {Align *} | v | _ {\ mit L ^ {P }}&:=\ left (\ int _ {\ Omega }\, \ left | V (x) \ right | \, ^ p \, DX \ right) ^ {1/p}, \, 1 \ Leq p \, <\ infty, \ | v | _ {\ mit L ^ {\ infty} &: = \ sup \ left \ {| V (x) | \ big | \,\, X \ In \ Omega \ right \} \ end {Align *} \]

• \ (\ Mit L ^ {2} (\ Omega) \) space is actually a Hilbert space assigned to the right inner product, \ (W, V) _ {\ mit L ^ {2} (\ Omega) }=\ int _ {\ Omega} \, w (x) \, V (x) \; DX \)
  • \ (| \ Cdot | _ 0 = | \ cdot | _ {\ mit L ^ {2} (\ Omega )}\)Remember
  • \ (\ Mit L ^ {p} (\ Omega) \) is a real-life (its dual space is \ (\ mit L ^ {q} (\ Omega )\), \ (\ frac {1} {p} + \ frac {1} {q} = 1 \), and only \ (\ mit L ^ {2 }\) it is a Hilbert space (the dual space is itself ).
  • \ (H \ ddot {o} lder? \) Inequality: \ (\ big | \ int _ {\ Omega} \, w (x) V (x) dx \ big | \ Leq \, | w | _ {\ mit L ^ {p} (\ Omega)} | v | _ {\ mit L ^ {q} (\ Omega )}, \, \, \ frac {1} {p} + \ frac {1} {q} = 1? \)
  • \ (\ Mit L ^ {p} (\ Omega) \) $ \ subset \ mit L ^ {q} (\ Omega), Q \ Leq P. $

• Distribution Function (\ (distributions \))

  • \ (\ Mit C ^ {\ infty} _ {0} (\ Omega) \) is \ (\ Omega \) with a compact set (I. e. bounded open set \ (\ Omega '\ subset \ Omega \), \ (D (\ partial \ Omega, \ Omega)> 0 \) infinite dimension micro-function functions exist, and any derivative of the Order on the boundary is zero. Sometimes it is recorded as \ (\ mathcal {d} (\ Omega )\).

  • \ (\ Mathcal {d} (\ Omega) \) derivative of the element \ (\ mathcal {d} ^ {\ Alpha} V: = \ frac {\ partial ^ {| \ Alpha |} V }{\ partial ^ {\ alpha_1} X_1 \ partial ^ {\ alpha_2} X_2 \ cdots \ partial ^ {\ alpha_d} x_d }, \) where \ (| \ Alpha | = \ alpha_1 + \ alpha_2 + \ cdots + \ alpha_d. \)

  • \ (V_n \ In \ mathcal {d} (\ Omega) \) converged on \ (V \ In \ mathcal {d} (\ Omega )\) it means that a bounded closed subset \ (k \) satisfies any one \ (n \), \ (v_n \) is 0 outside \ (k, and for any non-negative indicator \ (\ Alpha \), derivative \ (\ mathcal {d} ^ {\ Alpha} V \) consistent convergence on \ (\ mathcal {d} ^ {\ Alpha} v. \)

  • Distribution\ (\ Mathcal {d} (\ Omega) \) any element in the dual space is called a distribution, that is, the distribution is \ (\ mathcal {d} (\ Omega )\) the linear functional above, \ (L \ In \ mathcal {d'} (\ Omega) \) and \ (V \ In \ mathcal {d} (\ Omega ),\) \ (L (v) = <L, \, V> \) \ (dualiy \, \, pairing. \)

  • Define a norm of \ (\ mathcal {d} (\ Omega) \), \ (| v | _ k = \ sup _ {\ Omega} | V (X) |, \, V \ In \ mathcal {d} (\ Omega ). \) (you can prove it yourself)

  • $ \ Mit L ^ {p} (\ Omega) $ \ (\ subset \ mathcal {d'} (\ Omega )\), but $ \ mathcal {d'} (\ Omega) \ not \ subset \ mit L ^ {p} (\ Omega), p \ Ge, 1. $

    Step 1 proves that $ \ forall L \ In \ mit L ^ {p} (\ Omega) $ is a linear functional of \ (\ mathcal {d} (\ Omega;
    \ [\ Begin {Align *} l (v) & = <L, V >=\ int _ {\ Omega} l (x) V (x) \, dx, \ forall V \ In \ mathcal {d} (\ Omega ). \ L (\ alpha_1 v_1 + \ alpha_2 V_2) & = <L, \ alpha_1 v_1 + \ alpha_2 V_2 >\& = \ alpha_1 <L, v_1> + \ alpha_2 <L, v_2 >\&=\ alpha_1l (v_1) + \ alpha_2l (V_2), \,\,\,\,\,\,\forall \ alpha_1, \ alpha_2 \ In \ mathbb {r}, v_1, V_2 \ In \ mathcal {d} (\ Omega ). \ end {Align *} \]
    Step 2 proves that \ (L \) is continuous, that is, proof \ (| L (V) | \ Leq c | v | _ {\ mathcal {d} (\ Omega )}. \) Let's prove it:
    \ [\ Begin {Align *} l (v) & =\ int _ {\ Omega} l (x) V (x )\, DX \ Leq | L | _ p | v | _ Q \ & \ Leq | L | _ p (\ int _ {\ Omega} | V (x) | ^ {q} dx) ^ {\ frac {1} {q }}\& \ Leq | L | _ p | v | |_{\ mathcal {d} (\ Omega )} | \ Omega | ^ {\ frac {1} {q }}\& \ Leq c | V ||_{\ mathcal {d} (\ Omega )}. \ end {Align *} \]
    Step 3: Prove $ \ mathcal {d'} (\ Omega) \ not \ subset \ mit L ^ {p} (\ Omega), p \ Ge, 1. $

    ? For example, $ \ mathcal {d'} (\ Omega) \ subset \ mit L ^ {p} (\ Omega), $ is represented by \ (risze, \ (\ forall V \ In \ mathcal {d} (\ Omega )\), \ (U \ In \ mit {L ^ {p} _ {\ Omega }}\),

    ? The following formula is met:

Numerical Solution of partial differential equations-learning Summary