report2.ipynb

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    "Setup:\n",
    "Consider the 1-d boundary value problem from fluid dynamics:\n",
    "$-u''(x)+V(x)u'(x)=f(x),x \\in [0,1]$ <br>\n",
    "$u(0) = u(1) = 1$ <br>\n",
    "and we will set: <br>\n",
    "V(x) = $\\gamma$ <br>\n",
    "f(x) = 1\n",
    "\n",
    "Part 1: Problem formulation: <br>\n",
    "a: write this as a variational problem: $A(u,\\phi)=F(\\phi)$\n",
    "for some test function $\\phi$ <br>\n"
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   "source": [
    "$A(u,\\phi) = F(\\phi)$\n"
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   "source": [
    "Let $-V(x)=\\gamma$ and $f(x)=1$ $$-u^{''}(x)+V(x)u^{'}(x)=f(x),x\\in[0,1]$$ $$u(0)=u(1)=0$$\n",
    "$-u^{''}+ru^{'}=1$ and multiply through $\\phi$ we will get \n",
    "$$-\\phi*u^{\"}+\\phi*ru^{'}=\\phi$$\n",
    "Next we Integrate via the boundary conditions $[0,1]$ This will give $$ -\\int_{0}^{1} \\phi*u^{\"}dx+\\int_{0}^{1} \\gamma*\\phi*u^{'}dx=\\int_{0}^{1} \\phi dx$$"
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    "\\section*{Problem 1: Derive Weak Solution}\n",
    "Let $-V(x)=\\gamma$ and $f(x)=1$ $$-u^{''}(x)+V(x)u^{'}(x)=f(x),x\\in[0,1]$$ $$u(0)=u(1)=0$$\n",
    "$-u^{''}+ru^{'}=1$ and multiply through $\\phi$ we will get \n",
    "$$-\\phi*u^{\"}+\\phi*ru^{'}=\\phi$$\n",
    "Next we Integrate via the boundary conditions $[0,1]$ This will give $$ -\\int_{0}^{1} \\phi*u^{\"}dx+\\int_{0}^{1} \\gamma*\\phi*u^{'}dx=\\int_{0}^{1} \\phi dx$$ Let $n=2$ and Note the fact that $$u(x)=\\sum_{j=0}^{\\infty} C_{j}*\\phi(x)$$ Also the fact that $$u^{'}(x)=(\\sum_{j=0}^{\\infty} C_{j}*\\phi(x))^{'}=u^{'}(x)=\\sum_{j=0}^{\\infty} C_{j}*(\\phi(x))^{'}$$\n"
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    "$$\\int_{0}^{1} (\\phi_{i}(x))^{'}*\\sum_{j=0}^{\\infty} C_{j}*(\\phi_{j}(x))^{'}dx+\\gamma*\\int_{0}^{1} (\\phi_{i}(x))*\\sum_{j=0}^{\\infty} C_{j}*(\\phi_{j}(x))^{'}dx$$\n",
    "\t"
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    "$= -\\int_{0}^{1} \\phi_{j} dx$ We Can calculate factor out all the Cofficents of $C_{j}$ We get the following \t"
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    "$$\\sum_{j=0}^{\\infty} C_{j}(\\int_{0}^{1} (\\phi_{i}(x))^{'}*(\\phi_{j}(x))^{'}dx+\\gamma*\\int_{0}^{1} (\\phi_{i}(x))*(\\phi_{j}(x))^{'}dx))$$ "
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   "source": [
    "$= -\\int_{0}^{1} \\phi_{j} dx$ "
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    "We can define $x=\\sum_{j=0}^{\\infty} C_{j}$ and $A_{1}=(\\int_{0}^{1} (\\phi_{i}(x))^{'}*(\\phi_{j}(x))^{'}dx$ "
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    " and  $A_{2}=\\gamma*\\int_{0}^{1} (\\phi_{i}(x))*(\\phi_{j}(x))^{'}dx))$"
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   "source": [
    "and finally $$b_{j}=\\int_{0}^{1} \\phi_{j} dx$$ Let us sum up $A_{1} $ and $A_{2} $ to get $$A_{1}+A_{2} =A$$"
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    "\n",
    "b: Take $\\phi_i$ to be the hat functions discussed in class and aproximate u as a linear combination of these basis vectors: $u(x)=\\sum_i u_i \\phi_i(x)$. Show that part (a) becomes a linear algebra problem of the form Ax=b.\n",
    "Derive expressions for the matrix A and bector b.\n",
    "Show $A_ji = A(\\phi_j,\\phi_i)$ and can be writte as the sum of two matrices A = A<sub>1</sub> + A<sub>2</sub> where A<sub>1</sub> and A<sub>2</sub> corresopond to the first two terms on the left hand side of the ODE"
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    "c: Implement a routine that will return A as a sparse matrix and b given n and $\\gamma$"
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   "source": [
    "import MatGenerator as mg\n",
    "\n",
    "#print(mg.generateA(4,.3))\n",
    "print(mg.generateB(4))"
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    "Part 2: implement GMRES algorithm\n",
    "signature mygmres(l,b,x0,n,M,A)\n",
    "returns solution of Ax=b with initial iterate x0\n",
    "n is the dimension of the problem, A is nxn, M is nxn tha defines inner product use for calculateing vector norms (and error). Github, commits from all group members"
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    "Part 3: Solving the FEM problem <br>\n",
    "use GMRES to solve for cases V(x) = 1 and V(x) = n+1 <br>\n",
    "using M as the identity matrix. for each case, run with n=12,32,64,128 and l=2,4,8,16,32,64, increasing l until the error (norm of residual divided by n) is below 10<sup>-6</sup><br>\n",
    "plot the most accurate solution (as a function of x) as well as the error vs functions of n and l"
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