# Questions tagged [elliptic-pde]

For questions about solving elliptic PDEs, a special class of second-order linear PDE in two variables.

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### Simple to program method for elliptic PDE with curved boundary?

I know very little about numerical PDE, so I apologize if this equation is ill-formed (and I appreciate any corrections). I am currently learning about Brownian motion. A classic result is that we can ...
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### Elliptic PDE finite volume method with Dirichlet boundary condition

I want to discretize the following equation using a Finite Volume Method $$\nabla \cdot (a(x)\nabla u)=f(x)\\x\in \Omega \subset \mathbb{R}^2 \\u_{|\partial\Omega}=g$$ I'm using Voronoi cells here: ...
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### Finite element method for Surface integrals using polar coordinates

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 ...
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### 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 ...
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### 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 ...
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### Correctly setting boundary condition for periodic linear elasticity problem

From an old, wise engineering book Peterson's Stress Concentration Factors (http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0470048247.html page 324) I've got the following problem: There is 2D ...
1 vote
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### An example of mixed elliptic problem using lowest-order Raviart Thomas element

I try to solve the following mixed second order elliptic PDE in the domain $D=[0, 1]^2$ \begin{eqnarray*} v+\nabla p=&0 \quad &\text{in} \quad D,\\ \text{div}(v)=&1/2 \quad &\...
1 vote
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### How to correctly define the flux in a finite volume method for Poisson's equation with a piecewise constant material

How do we correctly define the flux in a finite volume method applied to Poisson's equation where we have a piecewise constant material? Specifically, say we have the equation \begin{align*} -\nabla\...
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I know that the piecewise linear finite element approximation $u_h$ of $$\Delta u(x)=f(x)\quad\text{in }U\\ u(x)=0\quad\text{on }\partial U$$ satisfies $$\|u-u_h\|_{H^1_0(U)}\leq Ch\|f\|_{L^2(U)} ... 4 votes 2 answers 124 views ### Three steps of pde numerical solution and nonlinear equation I'm trying to solve a nonlinear elliptic equation$$(n(u)u')' = f(u)$$and have a crucial misunderstanding. I suppose the procedure of solving some nonlinear equation consists of: Choosing a proper ... 1 vote 0 answers 59 views ### Decreasing - increasing - stabilising l_{2} norm Let \bar{x} denote the analytical solution of a PDE. Let x^{(k)} be the solution at the k^{th} iteration. The initial guess for the solution is x^{(0)} = 0. Let r_{0} = ||\bar{x}-x^{(0)}||_{2}... 2 votes 2 answers 320 views ### O(h^2) convergence for Elliptic PDE I am trying to solve an elliptic PDE in 2-D:$$-\nabla^{2} u = 20tanh(10x-5)(10-10tanh(10x-5)^2) = f$$I know that the solution is u = tanh(10x-5) but I am unable to get O(h^2) solution with a ... 1 vote 1 answer 109 views ### 3 questions on FEM to solve elliptic PDE with homogeneous and mixed boundary conditions Assume we use FEM with piecewise linear finite elements to discretize the BVP over \omega = (0,1): -u''+ bu' + u = 2x, u(0) = u(1) = 0 for parameter b\in R. Given a mesh T = \left\{x_i\right\}... 0 votes 0 answers 262 views ### How to form the stiffness matrix for the Poisson equation using a spectral method This is a follow-up question to How do I form the Chebyshev differentiation matrix in MATLAB? The goal of the following code is to solve the Poisson problem: ... 4 votes 1 answer 500 views ### Finite element error for second order ODE at nodes equal to zero I coded a finite element method with linear basis elements for the problem$$-u'' = f(x), x\in[0,1], u(0) = u(1) = 0 The nodes are uniformly spaced and I will denote them as $x_i$. I initially ...
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Let $\Omega\subset \mathbb{R}^d$, $d\in \{2,3\}$ be an open bounded polygonal/polyhedral set. Suppose I want to solve the following pde \begin{align*} \vec{q}+\vec{\nabla}u &=0\,&x&\in \...