# Questions tagged [finite-element]

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 estimates.

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### Does the time-dependent 1D advection-diffusion with point sources have an analytical solution?

I am looking for the analytical solution of 1-dimensional advection-diffusion equation with several point sources, Q, along the axial length of a cylinder through which the fluid flow occurs. Neumann ...
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### Bubble function property: how to prove it?

I'm studying error estimators and I need a check on an estimate. In our course, we've been given the following definition of bubble function (in 2D): It's a function defined on a triangle $T$ such ...
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### Computing the residual in a Dual Weighted Residual (DWR) method

I am in the process if computing the Dual-Weighted Residual (DWR) for a linear PDE with a linear functional but I am struggling with the residual part of the calculation. For example suppose we want ...
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### Displacement field not correct?

Consider the elastic equation $$- \operatorname{div}(C \nabla \mathbf{u}) = \mathbf{f}$$ as presented in step-8. Here $\mathbf{u}$ is the displacement vector, let's consider the 2d case. As you can ...
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### Confusion about bilinear form for elasticity equation in deal.ii tutorial

I'm learning how to solve vector-valued problems with deal.II library. In particular, I'm looking at the following introduction from the official website https://www.dealii.org/current/doxygen/deal.II/...
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### Dyadic operations, fourth order tensors and Tensor algebra

I am trying to understand the dyadic operation for a while since I am interested in Elasticity problems. I believe an intuitive understanding (rather than assuming) will give me good problem solving ...
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### Hanging nodes in deal.ii tutorials: how is the continuity constraint imposed?

While looking at step6 of deal.ii tutorials, I decided to try to understand how the constraints coming from hanging nodes are imposed. So I started by watching video lecture 16 by prof. Bangerth As ...
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### discretizing surface integral using nodal DG method

I am currently learning nodal DG methods, primarily through the book by Warburton, and am a bit confused on how to handle surface integrals using straight edged elements. On page 187 (and on page 214)...
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### Constrained optimization for non-linear equations in octaveGNU

I have installed Optim1.6.1 package. I would like to solve a system of equations in non linear finite element analysis using constraints as u=1 at certain nodes. u=0 at certain nodes. Typically I find ...
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### Reference for mass matrix assembly

I will now explain what I understand to be the process of a finite element mass matrix assembly. I would like a reference which does something similar, or if I am mistaken about the process please let ...
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### How to implement point source or volume source in finite element implementations

I'm trying to do a simple implementation to study the advection-diffusion-reaction dynamics in a straight pipe. I have points positioned along the length of the pipe (blue dots in the image above). I ...
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### RCM better than Nested dissection? (For FEM discretizations in 2D and 3D)

I realize this might be a too general question but here goes nothing: I am trying different re-ordering strategies and checking the fill-in of $A=LU$. I have 2D ($p=1$, $h=1/40$ on $\Omega = [-1,1]^2$)...
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### What is the weak form of a vector type Laplace equation?

For a scalar variable $u$, the weak form of the following Laplacian: $$\nabla^2 u =0$$ with the assumption that $v$ vanishes at the boundary is $$\int \nabla v . \nabla u \, d\Omega = 0$$ which is ...
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### Validity of algorithm for assembling the finite element global stiffness matrix

This blog post describes an algorithm for constructing the finite element global stiffness matrix as two basic steps: [Step 1] Initialize a table with the elements shared by a pair of connected node ...
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### Shape function for quarter point element from degenerate quadrilateral

I want an element as the one shown following. Nodes 6 and 8 are in the quarter position. Whether eight node quadrilateral element or six node triangle function can be used directly？ If not, how to ...
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### Manufactured solution for $-\operatorname{div}(a(x) \nabla{u}) = f$ when $\alpha(x)$ is discontinuous
I'm studying the dealii tutorial number 4,5 and I understand the workflow. I've also been able to find the EOC by using manufactured solution where $f$ is a smooth r.h.s. and $\alpha(x)$ smooth too. ...