Numerical Solution of partial differential equations-learning Summary

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 \ r $ is a bilinear ing that 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 \ 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 \ 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 \ 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 = \ r \), \ ({\ color {red} \ scr l (\ mit V; \ mathbf \ r )}\) is the dual space of \ (\ mit V \). It is often remembered 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 \ 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 \ infin ). \)

  • 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 \ infin ). \)

  • * 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 \ infin ).\)

    • \ (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 \ r ^ {d }\), \ (d \ Ge 1 \), and \ (\ Omega \) isLebesgueMeasurable.

  • \ [\ Begin {Align *} \ mit L ^ {p} &: = \ left \ {v \, \, \ big | \ int _ {\ Omega }\, \ left | V (x) \ right | \, ^ p \, DX \ Leq \ infin \ right \}, \, 1 \ Leq p \, <\ infin, \ mit L ^ {\ infin} &: = \ sup \ left \ {| V (x) | \ big | \,\, X \ In \ Omega \ right \} <\ infin. \ 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 \, <\ infin, \ | v | _ {\ mit L ^ {\ infin }&:=\ 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 $
  • \ ({\ Color {red} | \ cdot | _ 0 = | \ cdot | _ {\ mit L ^ {2} (\ Omega )}}\) \ ({\ color {black} {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 ^ {\ infin} _ {0} (\ Omega) \) is \ (\ Omega \) with a compact set (I. e. bounded open set \ (\ Omega '\ subset \ Omega \), \ (D (\ part \ Omega, \ Omega)> 0 \) infinite dimension micro function, 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 {\ part ^ {| \ Alpha |} v} {\ part ^ {\ alpha_1} X_1 \ part ^ {\ alpha_2} X_2 \ cdots \ part ^ {\ 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. \)

  • \ (\ Color {red} distribution \): \ (\ mathcal {d} (\ Omega) \) any element in the dual space is called a distribution, that is, the distribution is the linear function on \ (\ mathcal {d} (\ Omega) \), \ (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. $

    Proof: Step 1 proof $ \ forall L \ [\ In \ mit L ^ {p} (\ Omega) $ is on $ \ mathcal {d} (\ Omega) $ linear functional; \]
    \ 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 \ 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 )}. \)

    ? The following is a proof:
    $
    \ 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