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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437 views

Sequential approach to solving coupled PDEs

I'm dealing with a coupled system of three transient, non-linear convection-diffusion equations. Let's just say to simplify the problem that they take the following form: $$ -\nabla\cdot(D_{1}(u_{2},...
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1answer
480 views

DG local equation, how to interpret mean-averaged test function

In the paper http://www.sciencedirect.com/science/article/pii/S0045782509003521, an HDG element-local equation is described on page 584 equation (4), with one of the equations taking the following ...
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418 views

Simple turbulence model appropriate for buoyancy-driven cavity like problem

Which turbulence model is suitable for resolving incompressible buoyancy-driven flow of a fluid within an cylindrical ampoule? I prefer turbulence model which is sufficiently simple so that fully ...
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343 views

Energy conservation in the solution of the Helmholtz equation

This might be a silly question, but I know very little about the theoretical properties finite elements, so here goes. Suppose you were to solve the Helmholtz equation (let's say in 2D) with a ...
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262 views

What can be done with Finite Element Method and not with the Finite Volume Method, and vice versa?

What are some applications where you would absolutely go for either FEM, but not FVM, or vice versa? What are some applications where both methods are equally suited? I worked with the FEM so far and ...
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636 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 ...
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265 views

How to check if my stiffness matrix is correct

I built the stiffness matrix for the Poisson equation on a 2-dimensional domain with the shape of "almost" an octagon, using pyramid basis functions. I used almost to intend the fact that I have an "...
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634 views

Understanding Boundary Condition in FEM

I am trying to understand Dirichlet and Neumann boundary conditions in FEM and I wanted to know if my inference is correct. To articulate my understanding, lets consider a simple case of TE and TM ...
6
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138 views

Do practice and theory differ substantially when implementing Neumann Boundary Conditions using a Mixed Method?

I have implemented a pretty straightforward finite element solver for the following Poisson equation. For the purposes of this question we can assume the source term and the Dirichlet data both ...
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344 views

Will penalty-augmented stiffness matrix cause numerical issues in eigenvalue analysis?

In the finite element method, we often construct the constraints of the system by adding penalty-function terms ( which often are many many magnitudes, up to $10^6$ order bigger than the largest ...
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487 views

How to do FEM in sector elements?

Suppose we have an element in the polar coordinate as $0\leq r\leq 1$, $0\leq\theta\leq \alpha$. What are the correct basis functions in this element? For this kind of mesh, there are a lot of ...
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About the condition $\ker(B_h) \subset \ker(B)$ in mixed finite elements formulation

I'm studying mixed finite elements. The problem is a classical saddle-point one: we seek for $(u,p)$ in $V \times Q$: $$A u + B^t p = f$$ $$Bu = g$$ where $A: V \rightarrow V', B:V \rightarrow Q'$ ...
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84 views

Numerical methods for the continuity equation with Sobolev vector field

Consider the continuity equation $$ \partial_t \rho(x,t) + \operatorname{div} (b(x,t) \rho(x,t)) = 0, \qquad t \in [0,T], \quad x \in \mathbb R^N, $$ with $b \in L^1((0,T), W^{1,p}(\mathbb R^N))$. ...
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212 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{{\...
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260 views

Boundary conditions in the Finite Element Method at only one side of computational domain

I want to solve a Sturm-Liouville problem in 1D, i.e., \begin{align} [p(x)\ u'(x)]'+q(x)u(x) = f(x) \end{align} with boundary conditions \begin{align} u(0)=a \hspace{1cm} u'(0)=b \end{align} How do I ...
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308 views

Is it possible to predict the null space of a structure from contributing elements null spaces?

I am trying to solve an almost incompressible problem with heterogeneous properties by domain decomposition. Solution with CG converges slowly or divergerces completely. My problem becomes ill-...
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1k views

Poisson equation with pure Neumann boundary conditions (using FEM)

I am trying to derive the correct variational form for the Poisson equation with pure Neumann boundary conditions, and an additional contraint $\int_{\Omega} u \, {\rm d} x = 0$, as described in this ...
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214 views

Conjugate residual/gradient convergence checking in practice

Let's say we want to solve $Ax=b$ ($A$ symmetric positive /semi/definite) with the conjugate residual/gradient method. $A$ comes from FEM where the mesh is being refined. The exact solution is $x_*$ ...
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243 views

Negative viscosity stabilized by fourth order terms

I am trying to solve a "Navier-Stokes"-type problem where the viscosity is negative. Of course this renders the equation unstable and thus I add a fourth order term, so the entire equation becomes: $$...
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70 views

Global reconstruction defined elementwise in a-posteriori error estimator

This question is a follow-up of this previous one. In "Error Control for Discontinuous Galerkin Methods for First Order Hyperbolic Problems" by Georgoulis et al., an error estimator is ...
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179 views

How to apply an integrated constrain condition in FEM?

I'm running some simulation using FEM. In my model I need to apply a constraint condition to the governing equation. My governing equation similar to the diffusion equation as below: $$\frac{\...
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85 views

Extrapolation after successive finite element refinement

I wish to compute a certain function $\lambda$ (in my case an eigenvalue of the Laplacian) to a certain accuracy, which I wish to guarantee, if possible. I started with a finite element method. I know ...
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161 views

Geometric Multigrid for Conform and Non–″ Elements: Restriction Operators

First of all, let me set up some notations. Suppose we have a hierarchy of meshes $\mathcal{T}_0 \subset \mathcal{T}_1 \subset \dots \subset \mathcal{T}_n$. For simplicity I restrict myself here to $...
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202 views

Once and for all: Which FEM plattform should I use for a very large multiphysics simulation?

I'm fooling around with the decision on how to build a multiphysics simulation for too long now (also several questions in this forum). First, I thought it would be possible/necessary to write most of ...
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187 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 $$...
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55 views

Typo in a-priori error estimate in a Discontinuous Galerkin paper

I'm looking at this famous paper which is available in the link below: Franco Brezzi, LD Marini, Endre Süli, Discontinuous Galerkin methods for first-order hyperbolic problems, Mathematical Models ...
3
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1answer
149 views

Time discretization Navier Stokes equation

This question is a follow-up of this one. The weak form of Navier Stokes equation is (assuming $v,q$ test functions for the velocity and the pressure, respectively) $$(\frac{du}{dt},v)_{\Omega} + (\...
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117 views

How to treat nonlinear radiation term in heat equation using Finite-element method?

I am trying to solve the time-dependant non-linear heat equation with radiation. This equation is coupled to the radiative transfer equation but for the purpose of my question, this does not matter ...
3
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76 views

Jacobian Matrix of 2D element mapped to 3D

Note: I previously posted this question to MathStackExchange, but got no attention there. So I'm rewritting and trying over here. Problem summary Given a common¹ set of shape functions defined at ...
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80 views

Galerkin Least-Squares stabilization for FEM solution advection (hyperbolic) equations

I am playing with Galerkin Least-Squares stabilization to solve advection diffusion problem in the context of the finite element method. This works very well for steady-state advection-diffusion ...
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125 views

Construction of Prolongation and Restriction Operator for Geometric Multigrid (2D-FEM): Resulting in a Decreasing Solution

Consider the following problem, $$ -\Delta u(x) = f(x), \qquad x \in \Omega \\ u(x) = 0,\qquad x \in \partial \Omega$$ with $\Omega = [0,1]\times [0,1]$ being the domain and $\partial \Omega$ being ...
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55 views

Some formulations of domain coupling lead to saddle point problems. Is this merely an artifact?

Background I wanted to learn how to couple FEM and BEM (for the Poisson equation), because I wanted to better understand how open boundary conditions look like. Therefore I worked through the relevant ...
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80 views

How to construct a Fortin Operator for Crouzeix-Raviart Element?

I want to prove the LBB condition for the Stokes Equations discretised by the Crouzeix-Raviart element. The continuous Stokes Equation in the weak formulation is Find $u \in H_0^1(\Omega, \mathbb{R}^...
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1answer
87 views

Adaptive Lagrangian-Eulerian methods and practical benchmark results

Does anyone know of any published study that talks about the practical aspects of running Adaptive Lagrangian-Eulerian techniques for solid and/or fluid mechanics problems? I'm looking for things ...
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117 views

Solving saddle point problem having non-invertible top-left block with a PETSc nested matrix

My system is a symmetric FE problem with lagrange multipliers: $Z=\begin{pmatrix}A & C^T \\ C & 0\end{pmatrix}$ The matrix $A$ is positive semi-definite, non-invertible. The whole matrix is ...
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90 views

numerical instabilities in Fluid Dynamics, Finite Element Method

I'm looking for references to understand where the numerical instabilities come from in hydrodynamics in general, and notably when the Péclet number: $Pe>1$. I'm using the finite element method. ...
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87 views

Strain propagation from surface to bulk in COMSOL

I am trying to simulate strain propagation from the surface into the bulk. I have a rectangular semiconductor block (~2 μm thick) on top of which metal gates (~25 nm thick) are deposited as seen in ...
3
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89 views

Numerical solution to N-dimensional diffusion on simplex?

Assume I have a system of at least (but generally only) $N+1$ points in an $N$-dimensional space ($N > 3$ is possible). At each of these points $x_i, i=1,...,N+1$ I know an initial potential/...
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83 views

Structural Analysis Library

Can anyone recommend a structural analysis library that satisfies the following requirements: C++ API Simulate both beam elements and shell (slab) elements Both static and dynamic analysis Free and/...
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79 views

Element Preconditioner

Im just working on a preconditioner for the linear equation system $Ax = b$ arising in FEM for elliptic PDE. $A$ is a s.p.d Matrix with real valued entries. I read something about the element by ...
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68 views

Should I expect computational gains using a second-order splitting method here?

I am trying to solve a three-dimensional baroclinic transport problem. The hydrodynamic (three-dimensional shallow water) equations are: \begin{align} \nabla \cdot \vec{v} = 0, \tag{1} \\ \frac{\...
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0answers
80 views

Data transfer in the context of adaptive mesh in finite element method (FEM)

In the context of adaptive mesh in FEM, after a new mesh is created, the data on the old mesh are to be transferred to the new mesh. For the data on the integration points (IPs), it seems the usual ...
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220 views

Finite Element Stabilization for Drift-Diffusion/Advection-Diffusion Equations

I've tried my best to look through the relevant suggested similar questions when posting this, and hopefully this contains enough new material to not be considered a duplicate. I'm currently trying ...
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112 views

Piecewise Constant Enrichment of the continuous galerkin method

I am interested to study crack propagation in a hyperelastic material in a variational setting. The crack surface exhibits a jump discontinuity. The function space for displacement field should ...
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150 views

Alternating Schwarz does not converge without Dirichlet conditions on physical boundaries

Note: Thanks to comments, I realized that I have two problems, each which can be described more clearly on its own. This revised question covers the first. I would like to solve $$ -\Delta u - 1=0,\...
3
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0answers
718 views

1D Discontinuous Galerkin - Lagrange vs Legendre Basis

Consider the 1D advection equation in its strong and weak forms $$\ u_t + a u_x = 0 $$ $$\ \int_{x_{j-0.5}}^{x_{j+0.5}} w u_t \ dx - a \int_{x_{j-0.5}}^{x_{j+0.5}} w_x u \ dx + a [w(x)\hat{u}]_{...
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105 views

Spectral/hp-finite elements for 4th order PDEs

Does anyone know of references discussing the solution of 4th order PDEs by way of spectral/hp-finite element methods? Specifically, I'm interested in the extension of the spectral element method, ...
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188 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 ...
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136 views

Petrov-Galerkin enrichment method for Darcy equation

I was reading about Petrov-Galerkin Enrichment Method for Darcy equations. Here are a couple papers that discuss this in detail: A Petrov–Galerkin enriched method: A mass conservative finite element ...
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213 views

Stability analysis for explicit time discretization in the Finite Element Method

I have been looking for stability analysis of general reaction-diffusion problems, of the form $\frac{\partial u}{\partial t}=\nabla\cdot D\nabla u-k\,u$ , to be solved using the standard Finite ...

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