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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3 answers
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What are the conceptual differences between the finite element and finite volume method?

There is an obvious difference between finite difference and the finite volume method (moving from point definition of the equations to integral averages over cells). But I find FEM and FVM to be very ...
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51 votes
5 answers
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What are criteria to choose between finite-differences and finite-elements

I am used to thinking of finite-differences as a special case of finite-elements, on a very constrained grid. So what are the conditions on how to choose between Finite Difference Method (FDM) and ...
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33 votes
9 answers
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Modern resources for learning FEM

I need to get started using Finite Element Methods. I am about to start reading Numerical solutions of partial differential equations by the finite element method by Claes Johnson, but it's dated 1987....
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29 votes
9 answers
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What is a good way to run parameter studies in C++

The problem I'm currently working on a Finite Element Navier Stokes simulation and I would like to investigate the effects of a variety of parameters. Some parameters are specified in an input file ...
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27 votes
3 answers
5k views

What is the purpose of using integration by parts in deriving a weak form for FEM discretization?

When going from the strong form of a PDE to the FEM form it seems one should always do this by first stating the variational form. To do this you multiply the strong form by an element in some (...
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25 votes
4 answers
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How to incorporate the boundary conditions with the Galerkin method?

I've been reading some resources on the web about Galerkin methods to solve PDEs, but I'm not clear about something. The following is my own account of what I have understood. Consider the following ...
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24 votes
1 answer
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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?
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23 votes
9 answers
44k views

Basic explanation of shape function

I just started studying FEM in a more structured basis compared to what I used to do during my undergraduate courses. I am doing this because, despite the fact that I can use the "FEM" in commercial (...
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21 votes
1 answer
9k views

What is the general idea of Nitsche's method in numerical analysis?

I know that the Nitsche's method is a very attractive methods since it allows to take into account Dirichlet type boundary conditions or contact with friction boundary conditions in a weak way without ...
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17 votes
1 answer
6k views

How to derive the Weak Formulation of a Partial Differential Equation for Finite Element Method?

I have taken a basic introduction to Finite Element Method, which did not emphasize a sophisticated understanding of a 'weak formulation'. I understand that with the galerkin method, we multiply both ...
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16 votes
3 answers
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Finding which triangles points are in

Suppose I have a 2D mesh consisting of nonoverlapping triangles $\{T_k\}_{k=1}^N$, and a set of points $\{p_i\}_{i=1}^M \subset \cup_{k=1}^N T_K$. What is the best way to determine which triangle each ...
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15 votes
2 answers
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What is the purpose of the test function in Finite Element Analysis?

In the wave equation: $$c^2 \nabla \cdot \nabla u(x,t) - \frac{\partial^2 u(x,t)}{\partial t^2} = f(x,t)$$ Why do we first multiply by a test function $v(x,t)$ before integrating?
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2 answers
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FeniCS: Visualizing high order elements

I've just started messing around with FEniCS. I am solving Poisson with 3rd order elements and would like to visualize the results. However, when I use plot(u), the visualization is just a linear ...
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15 votes
1 answer
20k views

How to formulate lumped mass matrix in FEM

When solving time dependent PDE's using the finite element method, for example say the heat equation, if we use explicit time stepping then we have to solve a linear system because of the mass matrix. ...
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  • 1,859
15 votes
3 answers
1k views

Role of the numerical flux in DG-FEM

I am learning the theory behind DG-FEM methods using the Hesthaven/Warburton book and I am a bit confused about the role of the 'numerical flux.' I apologize if this is a basic question, but I have ...
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15 votes
1 answer
2k views

Visualizing discontinuous Galerkin/finite element data

I would like to visualize simulation results, obtained using the discontinuous Galerkin (DG) approach, within ParaView. Similarly to finite volume methods, the problem domain is divided into cube-...
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14 votes
2 answers
12k views

In FEM, why is the stiffness matrix positive definite?

In FEM classes, it's usually taken for granted that the stiffness matrix is positive definite, but I just can't understand why. Could anyone give some explanation? For instance, we can consider the ...
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14 votes
3 answers
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Alternatives to Comsol Multiphysics

This might be a question better suited for the Software Recommendations side of S.E., however I do believe that people who frequent this part of S.E. are more likely to be able to answer this question....
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14 votes
1 answer
447 views

What spatial discretizations work for incompressible flow with anisotropic boundary meshes?

High Reynolds number flows produce very thin boundary layers. If wall resolution is used in Large Eddy Simulation, the aspect ratio may be on the order of $10^6$. Many methods become unstable in this ...
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14 votes
2 answers
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Impose the compatibility conditions for mixed finite elements method in Stokes equation

$\newcommand{\v}[1]{\boldsymbol{#1}}$ Suppose we have following Stokes flow model equation: $$ \tag{1} \left\{ \begin{aligned} -\mathrm{div}(\nu \nabla \v{u}) + \nabla p &= \v{f} \\ \mathrm{div} \...
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14 votes
1 answer
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What are the relative benefits of using Adams-Moulton over Adams-Bashforth algorithm?

I am solving a system of two coupled PDE's in two spatial dimensions and in time computationally. Since the function evaluations are expensive, I would like to use a multistep method (initialised ...
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13 votes
3 answers
2k views

Mathematically, why does mass matrix / load vector lumping work?

I know that people often replace consistent mass matrices with lumped diagonal matrices. In the past, I've also implemented a code where the load vector is assembled in a lumped fashion rather than ...
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13 votes
6 answers
6k views

What is a common file/data format for a mesh (for FEM)?

I'm developing an FEM simulation. For early testing, I will use simple self-written mesher and visualisation of the mesh graph. But I want to prepare my program to use data generated by an existing ...
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13 votes
2 answers
2k views

Which preconditioners (and solver) in PETSc for indefinite symmetric systems should I use?

My system is a symmetric FE problem with lagrange multipliers (e.g. incompressible Stokes' flow): \begin{pmatrix}A & B^T \\ B & C\end{pmatrix} where $C = 0$ is the typical case (I have even ...
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13 votes
3 answers
971 views

Finite elements on manifold

I'd like to solve some PDEs on manifolds, say for example an elliptic equation on a sphere. Where do I start? I'd like to find something that use preexisting code/libraries in 2d , nothing so fancy (...
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13 votes
5 answers
2k views

Calculation of the sparsity structure for finite element matrices

Question: What methods are available to accurately and efficiently calculate the sparsity structure of a finite element matrix? Info: I'm working on a Poisson Pressure Equation solver, using Galerkin'...
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13 votes
1 answer
1k views

What are possible methods to solve compressible Euler equations

I would like to write my own solver for compressible Euler equations, and most importantly I want it to work robustly in all situations. I would like it to be FE based (DG is ok). What are the ...
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13 votes
2 answers
600 views

Verification in Eigenvalue problems

Let us start with a problem of the form $$(\mathcal{L} + k^2) u=0$$ with a set of given boundary conditions (Dirichlet, Neumann, Robin, Periodic, Bloch-Periodic). This corresponds with finding the ...
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13 votes
0 answers
453 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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12 votes
3 answers
390 views

How to write integration tests for numeric simulation software?

Just to be more precise, I'll put a worthy example of my typical use case. Let's say I'm developing a FEM software that produces several temporal solutions and inserts them in an HDF5 file, along with ...
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12 votes
1 answer
6k views

What is a "hanging node" in the finite element meshing?

When reading literatures about finite element method, the term "hanging nodes" can often be encountered. Could anyone tell me what indeed is a hanging node?
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12 votes
2 answers
410 views

Oscillations in singularly perturbed reaction-diffusion problems with finite elements

When FEM-discretizing and solving a reaction-diffusion problem, e.g., $$ - \varepsilon \Delta u + u = 1 \text{ on } \Omega\\ u = 0 \text{ on } \partial\Omega $$ with $0 < \varepsilon \ll 1$ (...
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12 votes
1 answer
354 views

How to integrate polynomial expression over 3D 4-node element?

I want to integrate a polynomial expression over a 4-node element in 3D. Several books on FEA cover the case where integrating is performed over an arbitrary flat 4-noned element. The usual procedure ...
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  • 501
12 votes
1 answer
518 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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11 votes
3 answers
4k views

Poisson equation: Impose full gradient as boundary condition via Lagrange multipliers

I have a physical problem governed by the Poisson equation in two dimensions $$ -\nabla^2 u = f(x,y), \; in \; \Omega $$ I have measurements of the two gradient components $\partial{u}/\partial{x}$ ...
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11 votes
2 answers
6k views

FEM: singularity of the stiffness matrix

I'm solving the differential equation $$ \left( \sigma^{2}(x) u ''(x) \right)'' = f(x), \;\;\; 0 \leqslant x \leqslant 1 $$ with initial conditions $u(0) = u(1) = 0$, $u''(...
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  • 445
11 votes
1 answer
4k views

Raviart-Thomas elements on reference square

I'd like to learn how the Raviart-Thomas (RT) element works. To that end I'd like to analytically describe how the basis functions look on the reference square. The goal here is not to implement it ...
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11 votes
3 answers
1k views

Best Methodologies for Managing a Mesh in Parallel Finite Element Computation?

I am currently developing a domain decomposition method for the solution of the scattering problem. Basically I am solving a system of Helmholtz BVPs iteratively. I discretize the equations using ...
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11 votes
2 answers
1k views

Space-time finite element discretization for time-dependent PDEs

In FEM literature, semi-variational methods are typically used in the solution of time-dependent PDEs. I have not seen a fully-variational approach i.e. where space and time are discretised by FEM, ...
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11 votes
1 answer
324 views

What are the possible numerical schemes for a diffusion equation with a nonlinear reaction term?

For some simple convex domain $\Omega$ in 2D, we have some $u(x)$ satisfying the following equation: $$ -\mathrm{div}(A\nabla u)+cu^n = f $$ with certain Dirichlet and/or Neumann boundary conditions....
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10 votes
3 answers
8k views

What is the difference between implicit FEM and explicit FEM?

What is the difference between explicit FEM and implicit FEM exactly? According to the post here, it seems that the only difference is whether implicit or explicit time integration is used. As I ...
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10 votes
2 answers
292 views

Finite elements $W^{1,\infty}$ error estimates

Are there finite element method setups that provide error estimates in the $W^{1,\infty}$ norm (i.e., bounds on $\|u'_h - u'\|_\infty$)? Which families of elements can be used for implementing them? (...
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10 votes
2 answers
5k views

How to efficiently implement Dirichlet boundary conditions in global sparse finite element stiffnes matrices

I am wondering how Dirichlet boundary conditions in global sparse finite element matrices are actually implemented efficiently. For example lets say that our global finite element matrix was: $$K = \...
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10 votes
2 answers
469 views

Why do the shape of finite elements matter?

I have used FEA for a couple of years now, but using it and using it correctly are two different things, safety factor is not the solution to everything. I have the feeling I won't be using it right ...
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10 votes
2 answers
503 views

FEM for vector valued problems: reference request

I'm a PhD working in a computational mechanics lab. I come from a Math department, and I have a good background for what concerns the basics of finite elements, like inf-sup conditions, DG, non-...
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10 votes
3 answers
561 views

Are there finite element software who handles more than five dimensions?

I'm a beginner with FE. My application is the pricing of financial derivatives where the space is five dimensional. So, adding time, the problem has six dimensions. I tried to look around (Fenics, ...
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10 votes
1 answer
995 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)} $...
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10 votes
4 answers
361 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 ...
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10 votes
2 answers
572 views

Which novel data structures are used in adaptive FEM?

A lot of adaptive FEM libraries use more advanced mesh data structures to handle adding/removing nodes, edges, triangles, tetrahedra, etc. For example, the p4est library uses octree data structures ...
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10 votes
1 answer
715 views

What is the impact of C++11 move semantics in the context of scientific computing?

C++11 introduces move semantics which can, for example, improve code performance in situations where C++03 would need to perform a copy construction or copy assignment. This article reports that ...
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