0
votes
1answer
42 views

Prove that $a(u-u_{h},u-u_{h})\ge 0$

Assume that $a$ is bilinear, symmetric and positive definite form, $u\in X$ and $u_{h}\in X_{h}\subset X$. I know the following fact: $$a(u-u_{h},u_{h})=0$$ Frm positive definiteness ...
14
votes
1answer
251 views

Fastest curve from $p_0$ to $p_1$

I'm trying to solve a problem in path planning: Given points $p_0$ and $p_1$ and vectors $v_0$ and $v_1$, find a function $p(t)$ st. $p(0) = p_0$, $p(T) = p_1$, $p'(0) = v_0$ and $p'(T) = ...
0
votes
1answer
98 views

Condition or Proof: Minimizer of one function is maximizing another function

I have two real functions $f(X),g(X)$ where the argument $X$ is a real matrix. The solution $X^*$ for the problem of minimizing $f$ is ending up maximizing $g$ as well. I am looking for a way to prove ...
1
vote
1answer
925 views

Derivative of $f(x,y)$ with respect to another function of two variables $k(x,y)$

Suppose that we have a function $f(x,y)$ of two variables: $$f(x,y) = g(x) + h(y) + 5(x-y) = x^2 + y^2 + 5(x-y)$$ where $g(x) = x^2$ and $h(y) = y^2$ are also functions of $x$ and $y$, respectively. ...
1
vote
0answers
90 views

Ritz method for the buckling of a plate using calculus of variation

I need to find buckling of plate. And i have got the constant in analytical solution which can be founded from variation methods. How can i find polynomial for the same function for Ritz method? I ...
1
vote
0answers
160 views

Finding a force function from bodies in equilibrium

(This is an edited version of the original question, since I'm starting a bounty) I'm trying to find a function $y$ from given data. Reverse optimization, so to speak. Say we have two ...
2
votes
1answer
234 views

How do I obtain an appropriate energy functional from the weak formulation of a partial differential equation?

I'm reading a textbook example on the finite element method: $\nabla^T[D(x,y,z)\nabla u] - a(x,y,z)u + f = 0 $ in R $\partial R= \partial R_1 \bigcup \partial R_2$, $\partial R_1 \bigcap ...
4
votes
1answer
193 views

Why weak formulations in numerical mathematics?

Regard the Poisson equation on the domain $\Omega = [-1,1]^n$ with $f \in H^{-1}$ $- \triangle u = f$ with homogenous Neumann boundary conditions. From standard regularity theory we know $u \in ...