Questions tagged [variational-calculus]

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6
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2answers
126 views

How to solve calculus of variations problems numerically?

For example, how to solve the well-known isoperimetric problem (i.e., to enclose the largest area with a fixed-length curve)? We can simplify things a bit and fix the two ends of the curve at $[a,0]$,...
0
votes
1answer
74 views

Deriving weak form of a set of scalar equations

I have the equilibrium equation in elasticity for a static case.i.e Div T=0. For certain implementation, I have to get the x and y component equations and then derive the weak form separately. How is ...
1
vote
0answers
97 views

functional second derivative

I'm trying to build a numerical solution for a parameter estimation problem of reaction-diffusion equation, using the adjoint method. To summarize it, I'm trying to minimize the function $$ G=\frac{...
1
vote
0answers
170 views

General-purpose Numerical Calculus of Variations Problem

Calculus of variations problems are generally cast in in the following simple form: find $u(t)$ that satisfies some boundary conditions and minimises $$ F[u] = \int_{t=0}^{t=t_f} f(u(t),u'(t),t) dt. $...
3
votes
1answer
154 views

Numerical computation of the velocity in the steady Navier-Stokes equation

I've asked this question on Math.SE too. Let $d\in\left\{1,\ldots,4\right\}$ $\Lambda\subseteq\mathbb R^d$ be bounded, nonempty and open and $\partial\Lambda$ be Lipschitz $V:=\left\{u\in H_0^1(\...
3
votes
0answers
167 views

Solving a nonlinear poisson equation via variational minimization

I am kind of new in finite elements and I am solving simple "Poisson nonlinear" problem. $- \nabla ((1 + u^2) \nabla u) = f$ $u = 0 \ \text{on} \ \Omega $ I am using Newton solver, where I have ...
2
votes
1answer
247 views

Finite element method applied to variational problem/functional VS weak formulation

I am confused. I read an introduction to finite element method where it was derived for the poisson equation: $$-\Delta u + cu = f,\qquad, u = g_0 \text{ on Dirichlet boundaries},\qquad\partial_n u ...
3
votes
0answers
152 views

Reference Request: Variational Problem

I want to solve approximately the following variational problem: Given a function $c:[-1,1]^2\rightarrow [0,1]$, constants $p_1...p_n\in \mathbb{R}^+$, $\alpha_1...\alpha_n\in \mathbb{R}$, and $\...