Discrete Maximum Principles

Consider the variational principle: \[ I(u)=\int _0^1 \left (u'(x)\right )^2+k^2\left (u(x)\right )^2 dx=\min \qquad \text {subject to}\qquad u(0)=u(1)=1. \] Suppose you want to use the Ritz Method to approximate the solution of this problem by a polynomial of degree \(n\). Derive a linear system for the coefficients of that polynomial.

Elliptic Problems:

Consider the standard five-point difference approximation (centered difference for both the gradient and divergence operators) for the variable coefficient Poisson equation \[ -\nabla \cdot (a\nabla v)=f \] with Dirichlet boundary conditions, in a two-dimensional rectangular region. We assume that \(a(x,y)\geq a_0>0\). The approximate solution \(\{u_{i,j}\}\) satisfies a linear system \(Au=b\).

1. State and prove the maximum principle for the numerical solution \(u_{i,j}\).

2. Derive the matrix \(A\) in the one-dimensional case and show that it is symmetric and positive definite.

3. For the one-dimensional and constant-coefficient case, show that the global error \(e_j=v(x_j)-u_j\) satisfies \(\|e\|_2=O(h^2)\) as the space step \(h\to 0\).

4. Discuss the advantages and disadvantages of trying to solve the system for the two-dimensional problem using (i) the SOR (Successive Over Relaxation) method and (ii) the (preconditioned) Conjugate Gradient method.

Elliptic Problems:

Consider the standard five-point difference approximation (centered difference for both the gradient and divergence operators) for the variable coefficient Poisson equation \[ -\nabla \cdot (a\nabla v)=f \] with Dirichlet boundary conditions, in a two-dimensional rectangular region. We assume that \(a(x,y)\geq a_0>0\). The approximate solution \(\{u_{i,j}\}\) satisfies a linear system \(Au=b\).

1. State and prove the maximum principle for the numerical solution \(u_{i,j}\).

2. Derive the matrix \(A\) in the one-dimensional case and show that it is symmetric and positive definite.

3. For the one-dimensional and constant-coefficient case, show that the global error \(e_j=v(x_j)-u_j\) satisfies \(\|e\|_2=O(h^2)\) as the space step \(h\to 0\).

4. Discuss the advantages and disadvantages of trying to solve the system for the two-dimensional problem using (i) the SOR (Successive Over Relaxation) method and (ii) the (preconditioned) Conjugate Gradient method.

Elliptic Problems:

Consider the standard five-point difference approximation (centered difference for both the gradient and divergence operators) for the variable coefficient Poisson equation \[ -\nabla \cdot (a\nabla v)=f \] with Dirichlet boundary conditions, in a two-dimensional rectangular region. We assume that \(a(x,y)\geq a_0>0\). The approximate solution \(\{u_{i,j}\}\) satisfies a linear system \(Au=b\).

1. State and prove the maximum principle for the numerical solution \(u_{i,j}\).

2. Derive the matrix \(A\) in the one-dimensional case and show that it is symmetric and positive definite.

3. For the one-dimensional and constant-coefficient case, show that the global error \(e_j=v(x_j)-u_j\) satisfies \(\|e\|_2=O(h^2)\) as the space step \(h\to 0\).

4. Discuss the advantages and disadvantages of trying to solve the system for the two-dimensional problem using (i) the SOR (Successive Over Relaxation) method and (ii) the (preconditioned) Conjugate Gradient method.

Wave Equation For the PDE \(v_t+cv_x=0\), derive the Lax-Wendroff scheme \[ u_j^{n+1}=u_j^n -\frac {1}{2}\left (\frac {c\Delta t}{\Delta x}\right ) (u_{j+1}^n-u_{j-1}^n) +\frac {1}{2}\left (\frac {c\Delta t}{\Delta x}\right )^2 (u_{j-1}^n-2u_j^n+u_{j+1}^n). \] Analyze this scheme in terms of accuracy and stability. Does this scheme satisfy a maximum principle? If it does, explain why. If it does not, give an example to show this.

Define and find the Green’s function for the discrete operator \(Lu=-\Delta u-\lambda u\) defined on the space of doubly infinite sequences \(u=\{u_n\}\), \(-\infty <n<\infty \), with \(\sum _{n=-\infty }^{\infty }u_n^2<\infty \), and with \((\Delta u)_n=u_{n+1}-2u_n+u_{n-1}\).

Newton Type Methods Suppose \(f(x)\) is a real-valued function defined for all values of the real variable \(x\), and that \(f'\) is Lipschitz continuous. Consider the discrete Newton method: Guess \(x_0\) and for \(k=0,1,2,\ldots \), do: \[ a_k=\frac {f(x_k+h_k)-f(x_k)}{h_k} \] \[ x_{k+1}=x_k-\frac {f(x_k)}{a_k} \] Analyze the local convergence properties of this method. How does the speed of convergence depend on the choice of \(h_k\) (ignore any effect of round-off errors in this analysis)? Now taking roundoff into account what would be a reasonable size for \(h_k\) in practice?