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# Questions tagged [elliptic-pde]

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### 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
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 ...
1answer
123 views

### FEM solution for Poisson is not exact at nodes

Let's consider the usual Poisson problem for FEM $$-u''(x)=1 \quad x \in [0,1]$$ with homogeneous Dirichlet boundary conditions. The solution is $u(x)=-\frac{1}{2}x(x-1)$ I know that the FEM solution (...
1answer
137 views

### C or fortran library to solve linear 2D/3D elliptic PDE

I am looking for a general purpose library which can solve a 2D or 3D linear elliptic PDE on a rectangular domain with mixed/Robin boundary conditions. I am a C programmer, so I would prefer a C ...
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
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 } \...
3answers
234 views

### Morley element implementation reference

I am looking for a detailed reference on the implementation of the Morley element for FEM, specifically for the biharmonic equation. By detailed, I mean that it should discuss the problems associated ...
0answers
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 ...
2answers
216 views

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2answers
88 views

### Elliptic equation with finite volume and unstructured high order geometry

I have found that in unstructured mesh, discretizing the laplacian operator with finite volumes requires special care, as given in An Introduction to Computational Fluid Dynamics: The Finite Volume ...
1answer
73 views

### Stabilization parameter for an elliptic equation

I simply want to solve the elliptic equation: $$-\kappa \nabla^2 u + u = f$$ where $f\in [0,1]$. When using continuous Galerkin with Lagrange elements, I have noticed that $\kappa$ has to be greater ...
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 ...
1answer
270 views

### Discontinuous Galerkin - Inhomogeneous Dirichlet B.C. for 1D Poisson Equation

I am trying to get some code working for the 1D Poisson equation using the textbook: Nodal Discontinuous Galerkin Methods Algorithms, Analysis, and Applications. I use the following formulation (for ...
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". ...
1answer
124 views

### Finite-volume method applied to a particular advection equation

I'm trying to apply the finite-volume method (FVM), with which I'm not so familiar, so a simple 1D PDE equation. I already asked this question in the math stackexchange, but was told that it could be ...
0answers
61 views

### scaling in discretized PDE system

I want to solve the following system via Matlab $\Omega=(0,1)^2$ $$\Delta y=\frac{1}{\alpha} p$$ $$-\Delta p= y -1$$ $$p|_{\partial \Omega}=0,~y|_{\partial \Omega}=0$$ using ...
1answer
257 views

### FEM current toy problem

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 ...
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\,\...
1answer
270 views

2answers
291 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 ...
1answer
103 views

2answers
154 views

### Stokes Equation in “two-fold saddle point” form?

Are there papers that deal with the (nondimensionalized) Stokes equation for incompressible fluid flow in a "doubly mixed" form like the following? \begin{align*} 0&=\underline{\epsilon} + \frac{...
1answer
99 views

### In mixed elliptic formulation, what are the weakest requirements to ensure the flux is in $H^1$?

In the book Mixed Finite Element Methods and Applications by Boffi, Brezzi, and Fortin there is a pretty long discussion about why the Raviart-Thomas (RT) projection is only defined for functions in \$...
1answer
411 views

### How do I avoid divide-by-zero when solving the Poisson equation with Fourier transforms?

I wanted to try to implement part of the method in the following article using Fourier transforms. http://www.shodor.org/media/content/jocse/student_submissions/nocito2010/nocito2010_pdf Right now I ...