A means of solving ordinary and partial differential equations. The domain of the problem is broken up into elements, and the solution in each element is expanded in a basis of functions. The Finite Element Method lends itself well to adaptive refinement, irregular geometry, and good error ...

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6
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1answer
40 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} + ...
0
votes
0answers
21 views

Is it reasonable to use B-splines to compute a radial symmetric electrostatic field?

When I use normal finite elements to compute the electrostatic potential, then the electric field itself is discontinuous at the interfaces of two cells. This impacts how electrons can be tracked ...
1
vote
3answers
115 views

Nodal basis functions and lagrange polynomials

I am a bit confused with nodal basis functions $h_i$ and the corresponding lagrange polynomials $l_i$. For linear functions, it's quite clear, on $[x_0,x_1]$ the nodal basis is $h_0 = l_0 = 1-x$. But ...
0
votes
1answer
76 views

Derivation in the FEM method

Can any on help me understand how from the simple delta function on the right hand and performing the integration we can get values as in the left? Note that $h = x_i + 1 - x_i$ The diffusion and ...
1
vote
2answers
57 views

The second variation of displacement interpolation function in Finite Element Method

I need to calculate the second variation of displacement interpolation function $u = \sum N_a u_a$ in Finite Element Analysis, where $N_a$ are the shape functions and $u_a$ are the nodal values. ...
4
votes
1answer
38 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 ...
1
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0answers
44 views

Discretization of lifting operator in BR2 scheme

The lifting operator $\mathbf{r}(\mathbf{v_h})$ for the '2nd version' of Bassi-Rebay scheme for elliptic problems in $d$ dimensions is defined as $$ \int_{\Omega_h} \mathbf{w}_h \cdot ...
0
votes
1answer
55 views

weak form of an equation by continuous discontinuous galerkin method

If we have the weak form of an equation like Poisson equation with discontinuous Galerkin method, to get the weak form of continuous discontinuous Galerkin method what terms are changed? Is it true ...
5
votes
0answers
75 views

Galerkin FEM error when using even number of elements

Intro: I am developing a Galerkin FEM code in matlab - starting small with a simple 1D ODE. The equation I'm trying to solve is: $$a u_x = cos(x), x \in [0, 2\pi]$$ Which has a known exact solution ...
0
votes
0answers
23 views

Estimate in Minimal Dissipation Local Discontinuous Galerkin Method

I am going through the paper titled "An analysis of the Minimal Dissipation Local Discontinuous Galerkin Method for Convection-Diffusion Problems", written by Cockburn and Dong. In section 3.1 of ...
1
vote
0answers
63 views

Solving a nonlinear poisson equation via variational minimization

I am kind of new in finite elements and I am solving simple "Poisson nonlinear" problem. $- \nabla ((1 + u^2) \nabla u) = f$ $u = 0 \ \text{on} \ \Omega $ I am using Newton solver, where I have ...
0
votes
1answer
44 views

How to find $L^2$ error for discontinuous Galerkin method

In standard finite element method, when we want to find $L^2$ error of a solution, we only find the error on Gaussian points that are inside of each triangle, for $DG$ method can we do this in a same ...
1
vote
1answer
67 views

How to construct shape functions in the $L^2(\Omega)$

It is necessary for me to find the shape functions on $L^2(\Omega)$(piecewise constant functions), I searched a lot but that, could not find anything, all of them are about higher degree polynomial. ...
4
votes
1answer
228 views

Intro to DG Finite Element methods

I wrote a number of 1D/2D FE and FD programs as a bachelor student, but the main problem I continually came into contact with was gradient shocks related to convection/diffusion problems in ...
0
votes
0answers
44 views

Check the well-posedness of the problem with level set equation

Given a domain $\Omega$ with 2 subdomains $\Omega_1,\Omega_2$ as figure below Level set equation $$ \partial_t\varphi + U\cdot\nabla\varphi =0 $$ $U$ is velocity only presents inside $\Omega_1$. ...
1
vote
0answers
64 views

CFD implementation in software [closed]

I've been learning CFD theory for 4 months (which includes FEA, FDM AND FVM) and now I want to start running simulations. As a novice to CFD software, I would appreciate some advice on where to start, ...
0
votes
0answers
130 views

how to solve a 2D non-linear Poisson equation?

I am trying to solve the following equation for $P(x,y)$: $$I = P \nabla \cdot \frac{1}{P^2} \nabla{P}$$ where $P$ and $I$ are functions of x and y. $I(x,y)$ and $P(x,y)$ I want to understand ...
1
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0answers
59 views

Is there a difference between the Galerkin MWR and other techniques called Finite Element Method [closed]

Recently, I took a course on numerical methods where we learned about Galerkin's Method of Weighted Residuals. We were told that it forms the basis of the Finite Element Method . However, in most ...
0
votes
1answer
37 views

Different Meshes Different Maximum velocities

I am working on a model in COMSOL and I am usin different meshes from coarse to fine keeping the other parameters constant. And, I got different maximum velocities for all the meshes. What can I do ...
1
vote
0answers
32 views

Representing a 3D system in 2D (Electromagnetic modelling)

Ok so I'm a complete beginner in computational modelling (I use analytical methods of physics typically) but I would like to model an anisotropic, aperiodic (but not random) finite array of metallic ...
0
votes
0answers
44 views

How to build semiconductor 1-D p-n diode model in COMSOL?

I don't have access to the semiconductors' module, thus I am building my own P-N model. I assumed there would be 4 intervals, ...
0
votes
1answer
67 views

How to create node to node lumping

I have been doing finite element analysis using Matlab. I look for many examples and tutorials producing only the stiffness matrix letting elements being weightless. However, in my case i need to do a ...
11
votes
1answer
142 views

What are differences between 'a priori' and 'posteriori' error estimate in numerical analysis?

I have learnt about Finite Element Method (also a little on other numerical methods) but I don't know what are exactly definition of these two errors and differences between them?
1
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0answers
100 views

Jump conditions on interior boundaries

I can not undrestand when we have jump on interior boundaries, if the solution be continuous. In the interior boundaries, derivatives have only different unit outward normal vector and this means we ...
4
votes
1answer
89 views

Simulate electric fields due to surface charges in simple circuits using python

I want to simulate the electric fields in simple circuits using Python and only free software. My first goal is to reproduce the images given in (1) which are made by the commercial ANSYS Maxwell ...
3
votes
3answers
101 views

How to assemble Global matrix (for coupled) problem?

I'm trying to assemble global matrices for the following system. $$ \begin{bmatrix} K& Q\\ Q^T&S \end{bmatrix} \begin{bmatrix} u_h\\ p_h \end{bmatrix} = \begin{bmatrix} f_u\\ f_p ...
1
vote
1answer
72 views

Finite element method applied to variational problem/functional VS weak formulation

I am confused. I read an introduction to finite element method where it was derived for the poisson equation: $$-\Delta u + cu = f,\qquad, u = g_0 \text{ on Dirichlet boundaries},\qquad\partial_n u ...
2
votes
2answers
166 views

FEM libraries with weak forms

I need to implement a structural analysis code, and I turn to you for advice. My needs are simple: Library must be integrable in a C++ code I want to express a weak form, without manual intervention ...
3
votes
1answer
105 views

Convergence of the second derivative of the finite element solution

Let $u_h$ be the finite element solution of a fourth order equation (like biharmonic equation), using polynomial degree two. If the convergence rate of $u_h$ is $2$, what is the convergence rate of ...
4
votes
2answers
130 views

FEM: Obtaining the Weak Form

In the the Finite Element Method (FEM), we attempt to obtain the Weak Form of the described equation. I understand that this is an attempt to reduce the order regularity of the equation, but what are ...
2
votes
3answers
81 views

Angle of rotation at a point in a deformed triangle

I have a 2D triangle which deforms with each vertex moving by some small ($\sin(x) \approx \tan(x) \approx x$) displacement vector. The displacement of any point in the triangle is linearly ...
1
vote
0answers
57 views

Solving system of equations with zeros on diagonal [closed]

I am using finite elements, Newton Raphson technique and MUMPS direct solver, to solve for $P$ in this equation: $$\frac{\delta K}{\delta x} \frac{\delta P}{\delta x}= \frac{\delta K B}{\delta x} + ...
3
votes
1answer
72 views

Continuous vs discontinuous pressure elements in fluid flow problems

When solving fluid flow problems using finite elements you typically end up requiring that $p\in L^2$. For Darcy flow problems, a popular choice of elements is the Raviart-Thomas element and ...
6
votes
1answer
52 views

Approximation properties of FEM projections operators on a boundary

We have an elliptic projection $$P: V \rightarrow V_{h}$$ which satisfies $$\Vert u - Pu \Vert_{L^{2}(\Omega_{e})} \leq Ch^{k+1} \enspace .$$ Can we say anything about $\Vert u - Pu \Vert_{L^{2}( ...
0
votes
1answer
64 views

How to deal with transition elements in adaptive fem

It is necessary for me to solve a Poisson equation by adaptive finite element method with transition elements technique to get conforming mesh. For the first local refinement everything is OK, ...
8
votes
5answers
498 views

Choice of basis in FEM

Briefly, what are the different types of basis used in FEM and why is the nodal basis so popular and advantageous in the finite element context?
2
votes
2answers
117 views

Finite element results by different meshes

There are some technique to generate mesh in a domain. My qustion is that: Is there any difference between the results using different techniques for mesh generation? If yes which one is better. For ...
3
votes
1answer
76 views

Gradients on 2D FEM Triangular Mesh

A scalar field f approximated on a triangular mesh using the FEM method can be given as $f(x,y) \approx \bar{f}(x,y) = \sum_{n=1}^3 \left(\phi_n(x,y) \cdot f_n \right)$ I am of the understanding, ...
1
vote
2answers
71 views

Simple question about find the element indicator

It is necessary for me to find the element indicator for poisson equation by linear basis in FEM. Therefore I have to find the following: $$\eta_k=h_k||f+\Delta ...
4
votes
1answer
87 views

Uniqueness of coefficients for shape functions of higher-order finite elements

In the FEM we usually use a trick to find the coefficients for shape functions: Finding $M^{-1}$ in the system $MC=I$, where for example in the linear case: $$ \begin{aligned} M&=[1 ,x_1,y_1;1 ...
0
votes
1answer
43 views

Comparing the following mesh grids for a star-shaped plate

Between these four different meshes for the same shape, compare them whilst considering the different results for displacement and stress. Moreover the difference in computational cost and accuracy of ...
1
vote
1answer
107 views

1D uniform flow test case for compressible flow

Are there any one dimensional hyperbolic test cases for a 1D uniform flow? (I need the test case for the purpose of showing the satisfaction of geometric conservation law on a moving mesh).
3
votes
1answer
107 views

a few questions on understanding geometric conservation law

I was wondering if anyone could help with understanding the geometric conservation law for moving domains. I came across Link1, and have tried to understood the paper by Farhat et al Link2. So far ...
1
vote
0answers
44 views

convergence of a method

I want to show convergence of a finite element method for a higher order equation.I have a coupled equations that solved together and gives two variables as answer $[u, v]$. $$w+\Delta u=0$$ ...
0
votes
1answer
67 views

Order of accuracy of DGFEM or FEM

I know that it is possible to determine the theoretical order of accuracy (B) of numerical solutions in FVM (for instance for a steady problem in which only the central differencing scheme (CDS) is ...
2
votes
1answer
58 views

Integrate over a face of a bubble function

I'm working in a finite element code where I need to calculate $$\int_F b_F$$ where $F$ is a face of some tetrahedron $K$ in the mesh, and $b_F$ is the usual bubble function defined by ...
7
votes
3answers
111 views

Reconstructing a continuous function from finite element method. Is there a faster algorithm for doing so?

Lets say I've decomposed a continuous function $y(x)$ over some domain $L_x$ using known finite element method with local basis $Q_i(x)$. Suppose $L$ is divided into $M$ "elements". If I want to know ...
11
votes
4answers
145 views

How to make a good mesh in a biologically accurate model with very small domains

I have been trying to make a biologically accurate 2D spatial model of tissue layers, where different physiological processes happen. This includes mainly chemical reactions, diffusion and fluxes over ...
1
vote
0answers
70 views

Numerical inversion involved confluent hypergeometric (1F1) (or Kummer function)

Edit: the question is solved ! Thank you for your time ! This problem arises when I tried to compute the valua of Asian call otions using Inverse Laplace transform method. Let $r=\mu = 0.15; \sigma ...
1
vote
0answers
18 views

Duality/Lagrangian condition and Variational Inequality of a cost functional

Given the functional $J_A(y,u) = \frac{1}{2}||y-y_d||^2_{L^2(w,\omega)} + \frac{\lambda}{2}||u||^2_{L^2(w^{-1}, \omega)}$ where w is a function belongs to Muckenhoupt class. Given the optimization ...