# Tag Info

Accepted

### Motivation behind Collocation Method

I would not insist on demanding a geometrical meaning from Galerkin methods in general. There is a connection, but it becomes less meaningful as you extend it further and further. (In a sense, it ...
• 12.3k
Accepted

### Don't we care about the numerical diffusion in the diffusion term?

We have the following problem: $$\frac{\partial u}{\partial t}+v\color{red}{\frac{\partial u}{\partial x}}-\nu\color{blue}{\frac{\partial^2u}{\partial x^2}}=0 \tag{*}$$ The function $u$ may represent ...
• 1,648
Accepted

### How do I integrate a function defined over an arbitrary area?

Instead of directly integrating over the area, it is often more convenient to use the divergence theorem to replace the area integral with an integral over the boundary edges. The divergence theorem ...
• 6,144

### Why do we have to resort to Higher order schemes for solving the 1-D advection equation/ continuity equation?

There is a difference between the requirements for a hyperbolic pde like $$u_t + a u_x = 0$$ and for a purely parabolic pde like $$u_t = u_{xx}$$ Suppose the solutions are smooth and you ...
• 3,028

### What is numerical damping in the context of time-dependent FEM solvers?

It is quite straightforward to demonstrate this for implicit Euler (aka backward Euler) on a scalar example. Consider the initial value problem $$\dot{y}(t) = i \alpha y(t), \ y(0) = 1$$ with solution ...
• 1,273
Accepted

### Residual norm of PDE discretization: correspondence in the continuous problem?

TLDR If you're using scalar products in FEM/FVM discretizations, use the mass-matrix scalar product, not $\ell_2.$ or If you're solving FEM/FVM systems with Krylov methods, precondition with the ...
• 3,126
Accepted

### Discretization with non-constant matrix containg entries form unknown vector

You can't. Nonlinear systems of equations are in general not solvable exactly. What you need to do is to use a method to solve nonlinear systems, of which there are of course quite a lot: A simple ...
• 55.8k
Accepted

### Projection method FVM poisson part, adding source term

The smooth solution turned out to have BC's applied in the following way: Walls and inlet: $\frac{\partial p}{\partial n}=0$ Outlet: $p=0$ Actually thought that we need only one value of P to pin, not ...
• 362
Accepted

### Time discretisation after splitting a 4th order equation

The introduction of $w$ is just a reformulation of your initial problem. If you use (2), this means that you make the 4th-order diffusion term explicit, which may potentially lead to stability issues. ...
• 1,943

### Finite difference discretization on a circle

The expressions you are using for eigenvalues and eigenfunctions are wrong (as per Wolfgang Bangerth's comment); therefore the results you are getting are not meaningful at all. There are analytic ...
• 8,692

### Is there Von Neumann stability analysis for 9-point laplacian like we have for the 5-point Laplacian?

There are actually a few different 9 point stencils in use, but they can all be written as a linear combination of the standard and skewed 5 point stencils. Performing the usual von Neumann analysis ...
• 2,249
Accepted

### Flux sign and face normal confusion in finite volume method

When dealing with conservation laws like your case, you can often make use of the divergence theorem (as you did). You can then express the fact that the total mass within your integration region is ...
• 2,985
Accepted

### Dividing a continuous domain into small squares; how to perform storage and querying?

After doing some quick research, I am convinced that the interviewer was looking for an answer related to one of the data trees. Depending on the application, one might be better than the others, but ...
• 2,821
Accepted

### Can I use Q0 finite elements when there are gradients involved?

Depends on the details. If you think about the traditional $H^1$ conforming formulation with the bilinear form $(\nabla \phi, \nabla v)$ it obviously cannot work because the stiffness matrix would be ...
• 2,104
Accepted

### constructing a symmetric matrix for finite difference

Here is an approach to achieve a symmetric finite difference operator for a space-dependent diffusion term $(r \psi_x)_x$. The trick is to apply central difference discretizations on first ...
• 1,114

### Discretization Error amplification instead of stagnation to machine precision

Yes. This behavior is to be expected and normal. When you are computing with a small value for $\mathrm dx$ then, to compute the difference quotient, you are subtracting two numbers that are nearly ...
• 558
Accepted

• 3,177
Accepted

### Discretizing the viscous component in 1 - D Navier stokes compressive flow

Consider a FV method as an approximation of the integral conservation law. Starting from the one-dimensional, scalar conservation equation $$u_t + f(u,\nabla u)_x =0,$$ ...
• 1,388

### Can I use Q0 finite elements when there are gradients involved?

There are two ways for what you are trying to do: If you approximate the gradient by finite differences involving the values at the cell midpoints, then your use of Q0 elements will lead to typical ...
• 55.8k