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# Questions tagged [elliptic-pde]

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### Increasing V-cycles for constant Coarsest Grid Size and increasing Fine Grid size

Problem statement I implemented geometric multigrid for $-\nabla^{2}=f$ where $f=\frac{3\pi^{2}}{4}sin \frac{\pi x}{2} sin \frac{\pi y}{2} sin \frac{\pi z}{2}$ on $\Omega \in [0,1]$ on a unit cube. ...
613 views

### Numerical implementation of the Dirichlet-to-Neumann map

I am solving the Dirichlet problem $$\begin{cases} \Delta u = 0, \\ u|_{\partial D} = f, \end{cases}$$ in a $2d$ domain $D$ using the finite element method. What I want to get is the ...
175 views

### How to compute numerical fluxes in the local discontinuous galerkin method for poisson equation 1D

Some days ago I began to study the local discontinuous galerkin (LDG) method, this is my first time working with a discontinuous method, so I decided to solve the poisson equation in 1D to learn the ...
201 views

Consider Poisson’s equation $$- \Delta u = f{\rm\qquad{ in }}\;\Omega$$ with following mixed boundary cconditions $$u = g{\rm\qquad{ on }}\;\Gamma \subset \partial \Omega$$ $$\frac{{\... 0answers 162 views ### Elliptical problem with Robin BC Working in finite differences, I am using a transformation on the temperature variable \Theta = \int_{T0}^T \kappa(T)dT to linearize the steady-state heat equation into a Poisson equation -\... 0answers 206 views ### Unstable convergence of a Poisson equation What could be the reason that the solution of a Poisson equation is smooth when obtained by an iterative solver, only if the maximum residual is set to a high value (e.g. 0.1)? When the maximum ... 0answers 73 views ### Solving numerically a linearized system of elliptic (?) Navier-Stokes equation (Shallow Water Derived) For my PhD Thesis, my advisor asked me to build a solver inspired from the article "Optimal Control Theory Applied to an Objective Analysis of a Tidal Current Mapping by HF Radar, J-L Devenon, 1989". ... 0answers 500 views ### Neumann-Neumann boundary intersection I am trying to solve a Vertex Centered, Finite Difference, Poisson equation -\nabla^{2}u=f with Dirichlet boundaries on the left and bottom and Neumann boundaries on the top and right using ... 1answer 440 views ### General case Kutta condition I'm working on a 2D inviscid fluid simulation using a "panel method", with Potential being used to enforce the no-through boundary condition. I'm trying to incorporate the Kutta condition, which says ... 0answers 345 views ### Solving PDE or eigenvalue problems without FEM Do you know any methods/solvers for PDE or eigenvalue problems like \begin{cases} \Delta u= 0\ (\text{ or }\lambda u) & \text{ in }\Omega \\ u =0 & \text{ on }\partial \Omega \end{cases} (... 0answers 185 views ### Are there known accuracy issues between 2D axisymmetric and 3D solutions? In my full 3D solutions I am solving for the potential throughout a 100\times 200\times 200 grid. Inside is a ring electrode set to -5V via a Dirichlet boundary condition, and surrounded on all ... 0answers 53 views ### Solving Laplace equation with constraint on boundary I have found the following PDE problem in a paper: Essentially, we have a rectangular domain where there is a unknown interface z=\xi(x) (liquid-air interface) separating the domain into two medium ... 0answers 47 views ### When to discretize nonlinear Poisson Equation I am trying to solve a nonlinear poisson equation of the form: u_{xx} + f(u_y)u_{yy} = 0. In trying to get a handle on this problem, it seems like there are two approaches. I could either ... 0answers 80 views ### Determine truncation error of PDE discretization The equation is$$\frac{\partial}{\partial x}\left(u\frac{\partial u}{\partial x}\right)=f(x)\\ 0<x<1, u(0)=u(1)=0$$I'm discretizing this PDE using FVM as follows: 0=x_0=x_{1/2}<x_1<x_{... 0answers 114 views ### PDE discretization on triangular domain Given the 2D Poisson equation$$\Delta u = f\\ u(x,0) = g_1(x), 0<x<1\\u(0,y) = g_2(y), 0<y<1\\ \partial_n u (x, 1-x) =0, 0<x<1$$defined on the domain \Omega := \{(x,y) \in \... 0answers 61 views ### scaling in discretized PDE system I want to solve the following system via Matlab \Omega=(0,1)^2$$\Delta y=\frac{1}{\alpha} p -\Delta p= y -1 p|_{\partial \Omega}=0,~y|_{\partial \Omega}=0using ... 0answers 199 views ### FiPy with derivative source terms I have a coupled nonlinear PDE system in 1 spatial dimension, in which I want to solve using FiPy. The dependent variables are n and T: \begin{align} \frac{\partial n}{\partial t} \,&=\, D\,\... 0answers 100 views ### Do DG methods for the Helmholtz equation always return positive quantities? Helmholtz Diffusion equation with reaction term: k\Delta u + u = f ~ \text{in} ~\Omega  \nabla u \cdot \mathbf{n} = 0 ~ \text{in} ~\partial \Omega $$For sufficiently small k (relative to ... 0answers 57 views ### Decreasing - increasing - stabilising l_{2} norm Let \bar{x} denote the analytical solution of a PDE. Let x^{(k)} be the solution at the k^{th} iteration. The initial guess for the solution is x^{(0)} = 0. Let r_{0} = ||\bar{x}-x^{(0)}||_{2}... 0answers 74 views ### Manufacturing a solution for non-smooth coefficients in elliptic problems This question is a continuation of this answer (I can't comment) If we were going to manufacture a solution for a problem with discontinuous coefficients, I understand that the solution should have ... 0answers 33 views ### Gradient convergence on a checkerboard domain in finite element I've got an elliptic problem in a domain with two materials in a checkerboard arrangement. Both materials are highly dissimilar, being one much more compliant than the other. If I use classic linear ... 0answers 121 views ### understanding interior penalty jump of basis function If cardinal basis functions are used (i.e. \psi_{ij}=1 iff i=j, and 0 otherwise) in interior penalty methods for elliptic equations such as SIPG & NIPG, shoudn't the jump in basis functions ... 0answers 577 views ### Finite Difference in Polar Co-ordinates Background Please note that I am duplicating the question on scicomp. I have already asked this in math. I am trying to come up with a scheme in Polar Co-ordinates for the following PDE: PDE I am ... 0answers 75 views ### 1D Poisson equation and quadratic basis functions assembly I'm solving the simple Poisson problem$$-u''(x)=1$$in the interval [0,1] with u(0)=u(1)=0. I discretised my domain as done here, i.e. with ... 0answers 59 views ### Difficulty to prove the coercivity of a(u, v)=\int_{\Omega} \nabla u \cdot \nabla v I'm stuck at proving the coercivity of the bilinear functional in the variational formulation of the problem: \begin{array}{c} -\nabla^{2} u=f \quad \text { in } \Omega \\ u=g_{D} \text { on } \... 0answers 53 views ### Existence and uniquness of solution of FVM for Poisson equation I'm discretizing the following Poisson equation using FVM where the domain \Omega of the solution is a regular hexagon of side 1 centered about the origin.$$\Delta u =k,\text{ $k$ constant}\\ \...
I am trying to solve a 2D elliptic PDE (see complete electrode model for electrical impedance tomography) using the finite element method (FEM) over a circular region $\Omega$. I have discretized the ...