Questions tagged [elliptic-pde]
For questions about solving elliptic PDEs, a special class of second-order linear PDE in two variables.
98
questions
3
votes
1
answer
148
views
How to compute numerically the $H^{1/2}$ norm of a function
I'm, in the context of FEM. Let's say I have a discrete function $g$ living on the boundary of my domain $D$. I need to compute numerically $||g||_{1/2,\partial D}$.
The definition I know is the ...
- 261
0
votes
2
answers
134
views
Practical implementation of the discrete compatibility condtion
I've recently started looking into writing a finite-differences-based solver for the Poisson equation of the form
$$\nabla\left(\varepsilon\nabla\varphi\right)=\rho$$ in 2D for arbitrary geometries (...
- 53
2
votes
2
answers
170
views
Poisson equation with discontinuous variable coefficient
Trying to solve numerically the 2D Poisson equation with variable diffusion coefficient $K$, that in general can be discontinuous.
$$ \frac{\partial}{\partial x} ( K(x,y)\frac{\partial C}{\partial x}) ...
- 435
3
votes
1
answer
162
views
ON the Kronecker product form of the laplacian matrix
It's well known that if we use 2nd order, centered, finite differences for the Laplace operator, we have that the matrix can be written as $$K=I \otimes A + A \otimes I $$ where $I$ is the identity ...
- 261
1
vote
1
answer
166
views
Gradient jump in weak formulation
I'm looking for an explanation of why the jump $[\nabla u]=[u] =0$ assuming $u \in H^2(\Omega)$.
We know that according to an embedding theorem, $H^1(\Omega)$ is a subspace of $C^0(\bar{\Omega})$ (...
- 25
2
votes
3
answers
123
views
Simple to program method for elliptic PDE with curved boundary?
I know very little about numerical PDE, so I apologize if this equation is ill-formed (and I appreciate any corrections).
I am currently learning about Brownian motion. A classic result is that we can ...
- 105
2
votes
1
answer
155
views
Index reduction of a DAE from a PDE system
I have a system of 2 non-linear, coupled PDEs that I would like to transform to a stiff ODE system to solve them using the method of lines. The 2 equations:
$$\begin{align}
\frac{\partial \phi}{\...
- 199
2
votes
2
answers
373
views
Semi-infinite domain transformation
Question is mostly related to literature or suggestions.
Given a semi infinite domain: $x=[0; +\infty);y=[0; +\infty)$. Willing to transform it to computational domain of: $[0,1]\times[0,1]$.
I did ...
- 364
3
votes
1
answer
92
views
"Optimal" domain partitioning in domain decomposition algorithms
When solving a PDE numerically by domain decomposition methods, what is the "optimal way" to split the domain? Are there any results stating that a particular partition of the domain yields &...
- 31
1
vote
0
answers
118
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 ...
- 217
0
votes
0
answers
245
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
...
- 261
1
vote
1
answer
262
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 (...
- 261
2
votes
1
answer
277
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 ...
- 1,038
1
vote
0
answers
55
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 ...
- 121
0
votes
0
answers
69
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 } \...
- 145
1
vote
3
answers
487
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 ...
- 971
2
votes
2
answers
293
views
Singularity in gradient caused by Dirichlet boundary condition
I am looking for a mathematical explanation for the singularity caused by a Dirichlet boundary condition partially imposed at a boundary.
For instance
$$
\nabla^2u=0 ~ \text{in}~\Omega
$$
where $\...
- 601
1
vote
2
answers
172
views
Simulating the heat equation with insulating material
My plan is to solve the heat equation in the right half portion of the domain, while having the left half completely isolated with constant temperature. To do so, I model the left half with a very low ...
- 601
1
vote
0
answers
104
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_{...
- 216
0
votes
0
answers
63
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.
$$\Delta u =k,\text{ $k$ constant}\\
\...
- 216
0
votes
1
answer
318
views
Elliptic PDE finite volume method with Dirichlet boundary condition
I want to discretize the following equation using a Finite Volume Method
$$\nabla \cdot (a(x)\nabla u)=f(x)\\x\in \Omega \subset \mathbb{R}^2
\\u_{|\partial\Omega}=g$$
I'm using Voronoi cells here: ...
- 216
0
votes
0
answers
161
views
Finite element method for Surface integrals using polar coordinates
I am trying to solve a 2D elliptic PDE (see complete electrode model for electrical impedance tomography) using the finite element method (FEM) over a circular region $\Omega$. I have discretized the ...
0
votes
1
answer
300
views
Imposing zero mean condition in FEM
I wanted to solve a periodic elliptic equation of the form $$-\nabla\cdot(A\nabla u)=-\nabla\cdot F$$ on $Y=[0,2\pi)^d$ using FreeFem++, where $A$ and $F$ are $Y$-periodic. The space of solutions is $...
1
vote
0
answers
153
views
PDE discretization on triangular domain
Given the 2D Poisson equation
$$\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$$
defined on the domain $\Omega := \{(x,y) \in \...
- 216
2
votes
0
answers
228
views
Are there known accuracy issues between 2D axisymmetric and 3D solutions?
In my full 3D solutions I am solving for the potential throughout a $100\times 200\times 200$ grid. Inside is a ring electrode set to -5V via a Dirichlet boundary condition, and surrounded on all ...
- 23
0
votes
1
answer
178
views
Elliptic PDE: Proving that a 2nd order accurate discretization of the 2nd derivative of the unknown is 2nd order accurate for the unknown itself
For an ODE:
$\frac{dy}{dt}=f(y(t),t)$
The Euler Explicit scheme reads:
$y_{n+1}=y_{n}+\Delta tf_n$
and it can be easily shown with a Taylor expension that:
$y_{n+1}=y_{n}+\Delta t \frac{dy}{dt}|...
- 435
6
votes
0
answers
278
views
$L^2$ norm error estimates of conforming FEM about Poisson’s equation with mixed boundary conditions
Consider Poisson’s equation
$$- \Delta u = f{\rm\qquad{ in }}\;\Omega $$
with following mixed boundary cconditions
$$u = g{\rm\qquad{ on }}\;\Gamma \subset \partial \Omega $$
$$\frac{{\...
- 61
2
votes
2
answers
92
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 ...
- 601
4
votes
1
answer
78
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 ...
- 601
3
votes
0
answers
270
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 ...
- 31
0
votes
1
answer
356
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 ...
- 1
3
votes
0
answers
92
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". ...
- 31
3
votes
1
answer
165
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 ...
- 45
1
vote
0
answers
65
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
...
- 111
3
votes
1
answer
296
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 ...
- 135
1
vote
0
answers
284
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\,\...
- 11
1
vote
1
answer
364
views
Raviart Thomas Mixed Finite Element with Mixed boundary conditions reference request
This is perhaps a more focused version of this question.
Using standard notation, I have a code that works and solves the following PDE using a Raviart Thomas mixed method.
$$\begin{align}
0 &= ...
- 1,000
3
votes
0
answers
712
views
Neumann-Neumann boundary intersection
I am trying to solve a Vertex Centered, Finite Difference, Poisson equation $-\nabla^{2}u=f$ with Dirichlet boundaries on the left and bottom and Neumann boundaries on the top and right using ...
- 467
0
votes
1
answer
201
views
Bifurcation of linear PDE
I have a linear elliptic PDE (unfortunately not allowed to be shown here) with a constant parameter $\epsilon$ giving the stable solutions qualitatively shown by the functions below.
As we smoothly ...
- 430
4
votes
1
answer
186
views
Reference Request: Raviart Thomas with hanging nodes
I am interested in reading about the analysis (existence, uniqueness, error estimates) of elliptic problems solved with a Mixed method that uses the Raviart Thomas elements (so far so good, easy to ...
- 41
1
vote
0
answers
108
views
Do DG methods for the Helmholtz equation always return positive quantities?
Helmholtz Diffusion equation with reaction term:
$$
k\Delta u + u = f ~ \text{in} ~\Omega
$$
$$
\nabla u \cdot \mathbf{n} = 0 ~ \text{in} ~\partial \Omega
$$
For sufficiently small $k$ (relative to ...
- 601
6
votes
3
answers
477
views
How to impose boundary condition with mixed derivatives?
I have the biharmonic equation on a 2D rectangular domain $\Omega$ with the following boundary conditions:
$\Delta^2 u = f$ on $\Omega$
$\nabla u \bullet \mathbf{n}=0$ on $\partial \Omega$ (1)
$u_{...
- 97
8
votes
0
answers
806
views
Numerical implementation of the Dirichlet-to-Neumann map
I am solving the Dirichlet problem
$$
\begin{cases}
\Delta u = 0, \\
u|_{\partial D} = f,
\end{cases}
$$
in a $2d$ domain $D$ using the finite element method. What I want to get is the ...
- 445
0
votes
2
answers
795
views
Correctly setting boundary condition for periodic linear elasticity problem
From an old, wise engineering book Peterson's Stress Concentration Factors (http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0470048247.html page 324) I've got the following problem:
There is 2D ...
- 193
1
vote
1
answer
323
views
An example of mixed elliptic problem using lowest-order Raviart Thomas element
I try to solve the following mixed second order elliptic PDE in the domain $D=[0, 1]^2$
\begin{eqnarray*}
v+\nabla p=&0 \quad &\text{in} \quad D,\\
\text{div}(v)=&1/2 \quad &\...
- 11
1
vote
1
answer
279
views
How to correctly define the flux in a finite volume method for Poisson's equation with a piecewise constant material
How do we correctly define the flux in a finite volume method applied to Poisson's equation where we have a piecewise constant material? Specifically, say we have the equation
\begin{align*}
-\nabla\...
- 747
10
votes
1
answer
1k
views
$L^2$-convergence of finite element method when right hand side is only in $H^{-1}$ (Poisson eqn)
I know that the piecewise linear finite element approximation $u_h$ of
$$
\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)}
$...
- 777
4
votes
2
answers
154
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
62
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}...
- 467
2
votes
2
answers
350
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 ...
- 467