DE Weak Form Linear System Two-Point Boundary Value Problem DE Weak Form 1 2 Linear System Discrete Form 4 3

spaces Definition: Definition: Remark: The space of all square integrable funcions defined in the domain Definition: The function and its first derivative are square integrable Remark: Both spaces are Hilbert spaces. R2 is also a Hilbert space R2 is also a Hilbert space with inner product

Triangle inequality Triangle inequality: Triangle inequality:

Cauchy-Schwarz inequality Cauchy-Schwarz inequality: (integral form) Example: verify CS-inequality

Cauchy-Schwarz inequality Is this true?: Cauchy-Schwarz inequality: (integral form)

Bilinear form Definition: Definition: Definition: Example: Example: The bilinear form is said to be symmetric if a(w, v) = a(v,w), ∀v,w ∈ V, A bilinear form on V is a function : V × V → R, which is linear in each argument separately Definition: the bilinear form a(・, ・) on V is bounded if there is a constant M such that. Example: Example: prove that a is bounded bilinear form on

Bilinear form Linear functional Definition: Definition: Example: the bilinear form a(・, ・) on is coercive if there is a constant α > 0 such that. A linear functional L : V → R is said to be bounded is the smallest constant c Example: prove that a is coercive on Remark: Example:

Lax-Milgram lemma 2 Lax-Milgram lemma Consider: where Hilbert space bilinear form on linear functional on Lax-Milgram lemma Hilbert space bounded coercive bilinear form on bounded linear functional on Then there exists a unique vector u ∈ V such that (2) is satisfied

Lax-Milgram lemma 1 2 2 1 DE Weak Form Example: Show that there exist a unique solution for (2)

Lax-Milgram lemma 2 1 Example: solution: Show that there exist a unique solution for (2) In order to show that there exist a unique solution for (2), we need to satisfy all the conditions of Lax-Milgram lemma solution: Poincare’s inequality (HW) proof Show that: Later we will do another proof for a symmetric a

Lax-Milgram lemma 2 1 3 Example: Show that there exist a unique solution for (3) 3 Thm: A finite dimensional subspace of a Hilbert space is Hilbert solution: In order to show that there exist a unique solution for (3), we need to satisfy all the conditions of Lax-Milgram lemma

Stability 2 Example: Definintion: Stability: A problem that satisfies the three conditions is said to be well posed 1)existence of solutions, 2)uniqueness of solutions, 3)stability 2 Poincare’s inequality Stability: continuous dependence of solutions with respect to perturbations of data Solution bounded by the data of the problem Setting ϕ = u in (2) and using (coercive) and (Poincare), we find Small change in the data produce small chang in the solution

Linear System of Equations 4 3 Remark: A is symmetric and positive definite Remark: Definition: Under what condition that (4) has solution An nxn matrix A is symmetric and positive definite if (4) has a unique solution iff that the matrix A is invertible ( non-singular ) Example: show that A is SPD

Linear System of Equations 4 3 Remark: A is symmetric and positive definite Remark: Proof: Under what condition that (4) has solution (4) has a unique solution iff that the matrix A is invertible ( non-singular ) coercively