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Having written this entry, let me also answer you question :-) The issue in essence boils down to the following: given a function $u(x)$ and its interpolation $u_h(x)$ onto either a triangular or quadrilateral mesh with the same number of unknowns, then which of the two is more accurate? In other words, is $$\| u - I_h^\text{triangles} u \| < \| u - ... 4 h is a measure of the mesh size. In the example, they are using rectangular elements. For which a commonly used measure of the mesh size is the length of the largest diagonal. Looking at the table above the sentence you quoted: (M_5-M_6)/(M_4-M_5)=(0.14052586 −0.14056422)/(0.14037251−0.14052586)\approx 1/4 = (h_6/h_5)^2 where M_i is the mean at i-th ... 4 The SEM is a FEM! It's almost like all these different names are designed to confuse the newcomer. I will speak primarily about the most popular form which uses a tensor product Lagrange basis with nodes at the Legendre-Gauss-Lobatto (LGL) points. Once you have chosen that basis and integrate with the LGL quadrature using the same nodes. You have a method ... 4 The main advantage is that it reduces the Runge phenomenon and leads to faster convergence rates. It also presents less numerical dispersion and need less nodes per wavelength (see 1 and 2). So, I would say that you would prefer the method for wave propagation scenarios. Regarding software that includes SEM, I am aware of the following: FSELib: Matlab ... 4 Specific answers to this question are probably time-limited. However, the following general approach (from the great Eric S. Raymond) works very well: Rule of Modularity: Write simple parts connected by clean interfaces. Rule of Clarity: Clarity is better than cleverness. Rule of Composition: Design programs to be connected to other programs. Rule of ... 4 Unfortunately there's no tool for this. You can run each on a variety of input sizes to establish the computational complexity they appear to have, i.e. the f in the O(f(n)) that characterizes each code. This can point you towards what underlying algorithms each is using and verify or not that something that should be O(n) is actually implemented to ... 3 I fully agree with Wolfgang's answer, but I'd like to add something of my experience. I had recently to choose between tri and quad based FEM and after googling for it, I decided that I would code both from scratch for a fair comparison (I had a previous experience in writing FEM kernel. I used C++ and the Eigen library for matrices and vectors. I considered ... 2 I will try answer based on my experience with deal.ii. The max_per_row=5 means that at most there will be 5 non-zeros per row in the matrix. Since we now know this then we do not need to have a 1\times{}100 matrix but rather a 1\times{}50. In other words this parameter sets an upper-bound on the memory needed. In reality it is not stored as one vector ... 2 You go the grouping in the term \nabla u^+-\nabla u^- mostly right but not quite. To see this, remember that the functions \phi_i^{\pm} are discontinuous, and that consequently u_0^+ is not necessarily equal to u_3^- -- if they were, you'd end up with a function that is continuous at that point. Instead, you need one degree of freedom per adjacent ... 1 You need to compute the projection \Pi(f-cU_h) as a first step. This is a finite element field, so you can create a global field (in deal.II lingo: Create the corresponding finite element and a DoFHandler object) and compute it by global projection by inverting a mass matrix. Alternatively, because the field is discontinuous, you can also forgo the global ... 1 As mentioned by @Wolfgang Bangerth , the value may be at the interior points. However, to visualise that better in paraview, You can use a filter called "plot over line" which plots the data points over a given line, which could come very handy for these kinds of clarifications in future. 1 @cfdlab's answer gives a general way to construct solutions for discontinuous coefficients, but here is a slightly more theoretical perspective to it as well. All of this can be understood in 1d, so imagine all of the functions below to be functions of just one argument x for a moment. First, you can't just choose any u and \alpha to obtain a function ... 1 If you want to use the coefficient of the form$$ \alpha = \begin{cases} \alpha_1 & r < 0.5 \\ \alpha_2 & r > 0.5 \end{cases} $$it is useful to work in polar coordinates. You build two pieces of the solution$$ u_1(r,\theta), \qquad r < 0.5 $$and$$ u_2(r,\theta), \qquad r > 0.5 $$Then at the interface you ensure solution and flux ... 1 This is not really a program as much as a big repository of codes. John Burkardt from Florida State University maintains a rich collection of scripts in C++, Matlab and Fortran for a large range of problems. You can check out the FEM 2D library or go up a level and look at other codes listed there. These have good commenting on them and you can look at the ... 1 If a homogeneous Neumann problem is considered, i.e. f=0 in \Omega and \mathbf{n}\cdot\nabla\phi on \Gamma, one solution is given by the constant function$$ \phi\left(\mathbf{x}\right)=\phi_0\,. $$Then the boundary integral equation reads$$\int\limits_{\Omega}\phi\left(\mathbf{x}' \right)\delta\left(\mathbf{x}'-\mathbf{x}\right)\mathrm{d}V' =\...