# Questions tagged [boundary-conditions]

For questions regarding the choice and/or appropriateness of conditions necessary to model a particular phenomenon with a partial differential equations.

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### Conservation of a physical quantity when using Neumann boundary conditions applied to the advection-diffusion equation

I don't understand the different behaviour of the advection-diffusion equation when I apply different boundary conditions. My motivation is the simulation of a real physical quantity (particle density)...
20k views

### Writing the Poisson equation finite-difference matrix with Neumann boundary conditions

I am interested in solving the Poisson equation using the finite-difference approach. I would like to better understand how to write the matrix equation with Neumann boundary conditions. Would someone ...
14k views

### How should boundary conditions be applied when using finite-volume method?

Following from my previous question I am trying to apply boundary conditions to this non-uniform finite volume mesh, I would like to apply a Robin type boundary condition to the l.h.s. of the domain (...
7k views

### Poisson equation finite-difference with pure Neumann boundary conditions

I'm trying to solve a 1D Poisson equation with pure Neumann boundary conditions. I've found many discussions of this problem, e.g. 1) Poisson equation with Neumann boundary conditions 2) Writing the ...
6k views

### How to properly apply non-homogeneous Dirichlet boundary conditions with FEM?

In general, Dirichlet boundary conditions won't be satisfied exactly for FEM for non-homogeneous boundary conditions. The FEM codes I've seen set the degrees of freedom to interpolate the Dirichlet ...
2k views

### Solving a linear equation system with pure Neumann condition

I am trying to solve a linear equation system $\textbf{A}\textbf{x}=\textbf{b}$, e.g. a Poisson equation discretized in strong form, using biCGstab method. Since there are only natural Neumann ...
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### What is the general idea of Nitsche's method in numerical analysis?

I know that the Nitsche's method is a very attractive methods since it allows to take into account Dirichlet type boundary conditions or contact with friction boundary conditions in a weak way without ...
12k views

### Periodic boundary condition for the heat equation in ]0,1[

Let us consider a smooth initial condition and the heat equation in one dimension : $$\partial_t u = \partial_{xx} u$$ in the open interval $]0,1[$, and let us assume that we want to solve it ...
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### How to incorporate the boundary conditions with the Galerkin method?

I've been reading some resources on the web about Galerkin methods to solve PDEs, but I'm not clear about something. The following is my own account of what I have understood. Consider the following ...
1k views

### Computing accurate fluxes with FEM

I have solved Poisson equation on a 3d domain with neumann and dirichlet boundary condition. I get the potential, take the gradient for each element and integrate on a surface of an element, I do this ...
4k views

### Applying Dirichlet boundary conditions to the Poisson equation with finite volume method

I would like to know how Dirichlet conditions are normally applied when using the finite volume method on a cell-centered non-uniform grid, My current implementation simply imposes the boundary ...
402 views

### Non-linear Boundary Value Problem. How to compute the Jacobian?

Consider a Boundary Value Problem: $$\delta u''+u(u'-1) =0 \Leftrightarrow u''=\frac{-u(u'-1)}{\delta}=:f(t,u',u), \\ u(0)=a, u(1)=b$$ $\delta,a,b$ are known parameters. I want to implement Newton'...
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### Discrete Poisson Equation with Pure Neumann Boundary Conditions

I'm trying to implement the Helmholtz-Hodge Decomposition in 2D, which states that a vector field is composed by a rotational free component, a divergence free component and a harmonic component. ...
620 views

### How can I evaluate more accurate energy eigenvalues from Schrodinger equation using shooting method?

I am trying to use the "shooting method" for solving Schrodinger's equation for a reasonably arbitrary potential in 1D. But the eigenvalues so evaluated in the case of potentials that do not have hard ...
7k views

### Boundary conditions for the advection equation discretized by a finite difference method

I am trying to find some resources to help explain how to choose boundary conditions when using finite difference methods to solve PDEs. The books and notes which I currently have access to all say ...
4k views

### FEniCS: separate boundary conditions in normal and tangential direction of mesh boundary

Given a vector-valued PDE, I'd like to enforce the boundary conditions $$\vec{n}\cdot u = g\\ \vec{n}\cdot \nabla (\vec{t}\cdot u) = 0$$ on the solution $\vec{u}$. If the boundary happens to align ...
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### Finite-volume method: can Dirichlet boundary conditions be applied to the integral form?

I would like to apply Dirichlet conditions to the advection-diffusion equation using the finite-volume method. This answer, "How should boundary conditions be applied when using finite-volume method?" ...
553 views

### Jump condition for elliptic equation in standard finite element method

Consider the elliptic PDE $-\mathrm{div}(k(x)\nabla u) = f(x) \in \Omega$ and $u = 0$ on $\partial \Omega$ If $k(x)$ is piecewise constant across an interface $\Gamma$ in the domain, then we ...
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### Dirichlet boundary condition for sparse matrix - Solving Ax=b only for free nodes?

I am solving Biot equation with sparse matrix in MATLAB. I have no problem with global sparse matrices assembly, but when I assign Dirichlet boundary condition, it is so slow. From this topic, How to ...
603 views

### Well-posedness of linear elasticity boundary conditions

I have several questions regarding suitable boundary conditions for linear elasticity. I have read that in order for modeling linear elasticity to be well-posed, the entire boundary cannot be ...
394 views

### Potential flow around a non-symmetric obstacle using stream functions

I've seen that there is a way to use the finite differences method, on a cartesian orthogonal grid, to perform calculations on potential flow about an obstacle without using the Neumann conditions, ...
270 views

### Do the class of PDEs that lack initial conditions have a name?

I am trying to think of what this kind of problem is called. I illustrate it with a telegrapher's equation with (hopefully) standard notation. Find $u:\Omega\times \mathbb{R} \to \mathbb{R}$ such ...
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### Do practice and theory differ substantially when implementing Neumann Boundary Conditions using a Mixed Method?

I have implemented a pretty straightforward finite element solver for the following Poisson equation. For the purposes of this question we can assume the source term and the Dirichlet data both ...
2k views

### Mixed boundary conditions Finite Element Method

I have the following problem in Finite Element Method $$-(\alpha u')' + \beta u' + \gamma u = f$$ with $\Omega = (0, 1)$, $u(0) = 0$ and $u'(1) = 3$ to be able to write the weak formulation ...
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### periodic boundary conditions for triclinic box

I am trying to do analysis on a data set of atomic coordinates generated form lammps. I simulated an alpha glycine crystal in a triclinic box. The box vectors look like the following, where xy,xz,and ...
938 views

### Solving Stokes flow with walls using Oseen tensor

Introduction I've developed a code to solve for generalised, incompressible 2D Stokes flow $\eta \nabla^2 \mathbf{v} - \nabla p + \mathbf{S} = 0$ $\nabla . \mathbf{v} = 0$ where $\mathbf{S}$ can ...
257 views

### How do I simulate an open end?

When simulating a partial differential equation describing a physical phenomenon like vibrations on a string, fluid flow in a chamber or quantum wave functions, the most straight-forward way is to ...
1k views

### Numerical method for a BVP with mixed boundary conditions (MATLAB)

I've been given a second-order non-linear ODE: $$\frac{d^{2}\theta(s)}{ds^{2}} = sf_{g}\cos{\theta} + sf_{x}\cos{\phi}\sin{\theta}$$ where $f_{g}, f_{x}$ and $\phi$ are constants. The boundary ...
640 views

### Neumann boundary conditions diffusion equations methods of lines

I want to solve the diffusion equation using the method of lines with Neumann boundary conditions  \frac{\partial p}{\partial t}=\frac{\partial^2p}{\partial x^2}\\ \frac{\partial p}{\partial x}(x=0)=...
848 views

### Flux boundary condition in solute transport

I have a pretty naive question, though important to me. Usually when solving the following PDE in solute transport: $\frac{{\partial C}}{\partial t } = \nabla. (D\nabla C -vC )=0,$ one can be asked to ...
248 views

### Well-posedness of Elasticity Boundary Conditions

For geotechnical engineering problems, it is common to fix a single component of displacement along a boundary as a Dirichlet boundary condition (roller boundary condition). However, I'm having ...
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### Schrodinger equation with periodic boundary conditions

I have a couple of questions regarding the following: I am trying to solve the Schrodinger equation in 1D using the crank nicolson discretization followed by inverting the resulting tridiagonal ...