28

Nitsche's method is related to discontinuous Galerkin methods (indeed, as Wolfgang points out, it is a precursor to these methods), and can be derived in a similar fashion. Let's consider the simplest problem, Poisson's equation: $$ \left\{\begin{aligned} -\Delta u &= f \qquad\text{on }\Omega,\\ u &= g \qquad\text{on }\partial\Omega. \end{aligned}\...


16

I think that one of your problems is that (as you observed in your comments) Neumann conditions are not the conditions you are looking for, in the sense that they do not imply the conservation of your quantity. To find the correct condition, rewrite your PDE as $$ \frac{\partial \phi}{\partial t} = \frac{\partial}{\partial x}\left( D\frac{\partial \phi}{\...


12

Sloede's response is very thorough and correct. I just wanted to add a few points to make it easier to grasp. Basically, any wave equation has an inherent wave speed and direction. For a one-dimensional wave equation: $$ u_t + a u_x = 0 $$ the wave speed is the constant $a$ which determines not only the speed at which the information is propagating in the ...


11

Curved boundaries are covered in most CFD books, e.g., Chapter 11 of Wesseling or Chapter 8 of Ferziger and Peric. While not a fundamental theoretical problem, the practical complexity of implementing boundary conditions for high-order methods on curved boundaries is a significant reason for interest in more geometrically-flexible methods such as the finite ...


11

In deal.II (http://www.dealii.org -- disclaimer: I'm one of the principal authors of that library), we do eliminate whole rows and columns, and it is not too expensive overall. The trick is to use the fact that the sparsity pattern is typically symmetric, so you know which rows you need to look into when eliminating a whole column. The better approach, in ...


10

From a numerical perspective, it's perhaps easiest to discuss the discretizations directly. For the Poisson equation with homogeneous Dirichlet boundary conditions, there is a unique solution for any right-hand side. Once discretized, the equation can be written in the form $Ax = b$, where $A$ is the standard discretization of the 3D Laplacian operator with ...


10

The problem you describe, how to prescribe non-reflecting or absorbing boundary conditions when solving partial differential equations (PDE) has been extensively studied. For complex (e.g. nonlinear) systems of PDE and general boundaries, it can be quite challenging and is something of an ongoing research problem. You can find many references on the topic, ...


9

This is rather a general remark on FVM than an answer to the concrete questions. And the message is that there shouldn't be the need for such an adhoc discretization of the boundary conditions. Unlike in FE- or FD-methods, where the starting point is a discrete ansatz for the solution, the FVM approach leaves the solution untouched (at first) but averages ...


9

The best way to do this is (as you said) to just use the definition of periodic boundary conditions and set up your equations correctly from the start using the fact that $u(0)=u(1)$. In fact, even more strongly, periodic boundary conditions identify $x=0$ with $x=1$. For this reason, you should only have one of these points in your solution domain. An open ...


9

The solution of the equation you are looking for is in the space $H^1$ of functions that have one weak derivative, but in 2d and 3d, this does not imply that the solution is in fact continuous. As a consequence, it is not possible to define the value of a solution at individual points, and equally consequentially, it is mathematically not possible to ...


8

This makes no sense. Let us assume that $g(z)=0$ has exactly one solution $z=z^\ast$, then your boundary condition $g(u|_\Gamma)=0$ is equivalent to $u|_\Gamma=z^\ast$, i.e., a linear Dirichlet condition. On the other hand, if there are multiple solutions of $g(z)=0$, then you are saying that the value $u|_\Gamma$ could have multiple values, but this does ...


8

Answering your last question first, do people actually use FDM for curved boundary nowadays I'd say the answer is no. In the commercial CFD world, 2nd order accurate finite volume schemes are the de-facto industry standard. One of the advantages of FV (and finite element/discontinuous galerkin approaches Jed mentioned) over FD is the much more natural ...


8

For the sake of simplicity let me slightly change your notation. Let $u, v, w$ be the components in the $x, y, z$ directions of a true (polar) vector, and $y=0$ the symmetry plane. For a true vector, symmetry conditions across the $xz$ plane are: \begin{eqnarray} u(x,y,z) &= u(x, -y, z) \\ -v(x,y,z) &= v(x, -y, z) \\ w(x,y,z) &= w(x,-y,z) \end{...


8

It is standard procedure for general case like that. You can enforce condition for electrode by Lagrange multipliers, $$ L(u,\lambda) = \int_\Omega \sigma u_{,i} u_{,i} \, \textrm{d}V - \int_{\partial\Omega_\sigma} u n_i j_i \textrm{d}S + \lambda\left(\int_E \sigma u_{,n}\, \textrm{d}S - I \right) $$ that for discretised problem is $$ \left[ \begin{array}{...


7

I think you are on the right way. If you correct your errors, it will look very similar to http://www.math.toronto.edu/mpugh/Teaching/Mat1062/notes2.pdf.


7

This is the wrong boundary condition. If your solution is smooth, then the correct condition at $r=0$ is $$ \frac{du}{dr} = 0. $$ You can see this by thinking about what would happen if you cut a line through the origin through the entire domain, i.e., you look at the solution not only for $r\ge 0$ but also for $r\le 0$. To the left of $r=0$ you of course ...


7

There is mathematical justification for setting Dirichlet boundary degrees of freedom to a value. However, you should adjust your variational form accordingly. If you are looking at a general problem, say: Find $u\in\mathcal{U}$ such that $a(u,w)=l(w) \ \ \forall w\in\mathcal{V}$ where $\mathcal{U}=\{u:\int \nabla u^2 < \infty, u=g\text{ on }\...


7

You can use a Nitsche-type method for this. See the following reference: J. Freund, R. Stenberg. On weakly imposed boundary conditions for second order problems. Proceedings of the Ninth Int. Conf. Finite Elements in Fluids, Venice 1995. M. Morandi Cecchi et al., Eds. pp. 327-336. I have implemented this a while ago in some simple FEniCS-Code to deal with ...


7

Typically, you would decompose your solution into $u = u_D + u_0$, where $u_D$ satisfies inhomogeneous Dirichlet conditions. You can then solve for $u_0 \in H^1_0$ subject to $$\Delta u_0 = f-\Delta u_D.$$ You would then recover a variational formulation over $H^1_0$ again. Note that $u_D$ is non-unique. To remedy this in proofs, $u_D$ is often taken to be ...


7

Kirill's answer is a good method but it is hampered that the fact that you need to tune it to a specific range in energy, so it is not appropriate if you have a broad-band wavepacket you need to absorb. An alternative approach is exterior complex scaling, which is reviewed well in Infinite-range exterior complex scaling as a perfect absorber in time-...


7

There is no shortcut. Just like there is no shortcut to becoming an "engineering expert in a short time". The thing is that to be an expert in civil engineering, you need to understand load analysis, use cases of buildings and bridges, materials, designs, and regulatory issues. For computational science, you need to understand the mathematical background, ...


6

I know its been 2 years.. but here is an example from physics to hammer it home. Consider you are solving the Poisson equation: $$\nabla^2\phi =f$$for electrostatic potentials(ϕ) with the right hand side: $$ f=(-ρ/εo) $$ And, ρ(the charge density) is specified at all the grid points. Statement: For Periodic solution, the ∫ρdv=0 condition (described by Ben ...


6

I don't have electrostatics training but your second picture looks OK to me. Your conductive inclusion is essentially shorting out the plates u1 and u2. I was taught that there is no electric field in conductive bodies, and therefore electrostatic potential is constant there. If the conductive body touches both plates u1 and u2 which are forced to ...


6

Let me give a part of your answer, I would need some more indications from your side to answer you fully. So please read, and write some comments so that I can complete my answer. About notations in numerical methods There are a few mistakes in the way you write your equations. These are only details, but for someone who is used to it, it can a bit ...


6

Preliminary remarks: The natural jump condition is on the normal derivative, not on the full gradient. The fact that the solution is continuous across the interface is part of the formulation of the problem. This is not only due to the finite element formulation. Formulation The standard formulation simply enforces it with... no extra term. Start by ...


6

There are multiple ways of implementing Dirichlet boundary conditions when using the finite element method. For each unknown $i$ of the system belonging to the Dirichlet boundary, you can zero out row $i$ of the Jacobian and entry $i$ of the right-hand side. At that rate, you can reformulate the problem for just the unknowns in the interior, which reduces ...


6

$K u$ equals the internal forces only in the linear case. The tangent stiffness matrix, $K$, in a nonlinear problem is normally used in a Newton-Raphson algorithm to calculate updates to the displacement vector as follows: $$K \Delta u = f - f_{internal}$$ $$ u_{i+1} = u_i + \Delta u$$ The vector of internal forces, $f_{internal}$ must be calculated from ...


6

From Ablowitz and Zeppetella we know that the analytical solution reads: \begin{equation} u(x,t)=\frac{1}{1+e^{{(-\frac{5}{6} t+ \frac{\sqrt{6}x}{6}})^2}} \end{equation} Usually, analytical solutions require the imposition of boundary conditions. Are there any used in the derivation of this expression? If so, you must use the same boundary conditions ...


6

The problem with ABC in the form of derivative operators at the boundary is that the exact ABCs are non-local in space and often in time, which makes them hard to use in numerical simulations. You can derive approximate ABCS based on series expansions, but this can be tedious and it may be difficult to get stable schemes. This approach was made famous by a ...


6

Such problems (sometimes called lateral Cauchy problems) are in general not well-posed (meaning they either lack a solution, or there are infinitely many of them, or the solution is unstable under perturbations of the boundary conditions). For parabolic (or dissipative) equations, it makes sense to study the stationary limit (simply omit the term $u_t$ in ...


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