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4

You would need to linearize the problem. I prefer to do it before discretization but it's possible to do also after discretization. (I'm a bit skeptical of linearization after discretization because I have never looked into the details. In general, discretization and linearization steps do not commute.) In the following I assume that the equation is actually ...


3

In the Navier-Stokes equations, you can't prescribe the pressure on the boundary (or part of it). That's just not a physical thing, nor mathematically correct. The only thing you can prescribe is the traction, i.e., the normal component of the stress, which is given by $$ \mathbf t = (-\nu \nabla \mathbf u + pI) \mathbf n. $$ For example, you could ...


3

What you are looking for is a Discrete Kirchhoff Quadrilateral plate or DKQ plate. Seems you are looking for a very straight forward formulation that simply give you the global stiffness matrix. But i'm afraid that most codes I've seen are dealing with integration and transformation. You can search for DKQ source code. There are documents for java which ...


2

For the case of faceted triangle geometry, the derivatives you're looking for ($\frac{d\phi}{dx}$, $\frac{d\phi}{dy}$, $\frac{d\phi}{dz}$) can actually be found without resorting to calculus (chain rule / jacobian), you can deduce them from purely geometrical considerations. These derivatives are the cartesian (x,y,z) components of the vector function $\...


2

Comments above are right: it seems that you are also integrating in time (and indeed you also set the numer of points in time in your code), but the equation is only in variable $x$. The following snippet produce the correct solution to your problem with linear elements in Python. To compute $\int_0^1 \phi_i(x) f(x)dx$ I used integrate.quad from scipy, ...


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As the mathematica.se thread shows, the solution of $$ \begin{aligned}\frac{\partial}{\partial x}\left( \operatorname{sign}(x) u(x) \right) + \frac{\partial}{\partial x} \left( \sqrt{u(x)} \frac{\partial u}{\partial x}(x) \right) &= 0 & &\text{in } \Omega = (-6,6), \\ u &= 0 & &\text{on } \partial \Omega = \{-6,6\} \end{aligned}$$ is ...


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First off, I'd note that your initial condition doesn't satisfy the boundary conditions, so you might want to instead use $u_0(x) = e^{-x^2} - e^{-L^2}$. A great sanity check for problems like yours is the conservation property -- the total mass of $u$ should stay the same. $$\begin{align} \frac{d}{dt}\int_{-L}^Lu\, dx & = \int_{-L}^L\frac{\partial u}{\...


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Since your trial and test spaces are different, you have to use a different version of Lax-Milgram lemma, see e.g., [1], Theorem 5.1.2 You can still use lifting idea since the PDE is linear. Then you can verify the conditions in standard Lax-Milgram lemma. To show coercivity, you need the condition $$ \gamma(x) - \frac{1}{2} b'(x) \ge -\eta, \qquad -\infty &...


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