Consider the initial boundary value problem

1
Consider the initial boundary value problem
Main idea
We approximate the function
by
For each fixed t,
Is a pc-wise linear function of x over a
triangulation
Weak Formulation
2
Weak Formulation
Discrete Equation
Semidiscrete problem
3
Discrete Equation
Matrix Form
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Matrix Form
stiffness matrix
Mass matrix
System of ODE has a unique solution
for each t
5
Stability condition of semidescrete
Set
Integrate
It means ???????
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Error
Weak Formulation
Discrete Equation
Theorem 10.1
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Theorem 10.1
Theorem 10.2