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1500 questions
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3 answers

Integrating Lagrange polynomials with many nodes, round-off

Given a set of points $\{x_j\}_{j=1}^n$ in $[-1, 1]$, I would like to compute $$ \int_{-1}^{1} L_i(x)\,\text{d} x $$ exactly. $L_i$ is the Lagrange polynomial with respect to the points $x_j$ with $x_i$ as node, i.e., $$ L_i(x) = \prod_{j\neq i}…
Nico Schlömer
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Estimate information entropy through Monte Carlo sampling

I am looking for methods that allow estimating the information entropy of a distribution when the only practical ways of sampling from that distribution are Monte Carlo methods. My problem is not unlike the standard Ising model that is typically…
Charles Wells
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$L^2$-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 $$ \|u-u_h\|_{H^1_0(U)}\leq Ch\|f\|_{L^2(U)} $$ provided that $U$ is smooth enough and $f\in…
Bananach
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CFL condition in Discontinuous Galerkin schemes

I have implemented an ADER-Discontinuous Galerkin scheme for the resolution of linear systems of conservation laws of the type of $\partial_t U + A \partial_x U + B \partial_y U=0 $ and observed that the CFL condition is very restrictive. In the…
Adr
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Solving a difficult system of equations numerically

I have a system of $n$ non-linear equations that I want to solve numerically: $$\mathbf{f}(\mathbf{x})=\mathbf{a}$$ $$\mathbf{f}=(f_1,\dots,f_n)\quad\mathbf{x}=(x_1,\dots,x_n)$$ This system has a number of characteristics that makes it particularly…
Szabolcs
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Are there any "light-weight" FEM packages around?

Basically, FEM seems to be a problem that is pretty much "solved". There are numerous powerful frameworks existing, like Trilinos, PETSc, FEniCS, Libmesh or MOOSE. One thing they have in common: They are extremely "heavy-weight". First, the…
Michael
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3 answers

The real myth of GPU (specifically CUDA) really speed up FEM/CFD

Now I have been believing that FEM/CFD is supposed to be faster on a GPU unit - here I am using CUDA as solid example. However, I have not been able to find a convincing paper where the benchmark actually appear to me that 'Yes, this is true!'.…
Quang Thinh Ha
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Numerical integration using interval arithmetic, nowadays

Is there now a package for rigorous numerical integration that uses interval arithmetic and has access to a well-developed library of special functions? By "well-developed", I mean something that, at the very least, includes the error function…
H A Helfgott
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3 answers

How to build a recursive spline function in C++

At the moment I am working on a differential equation solving method called basis-spline collocation. What I am having trouble with is building a method to build an arbitrary order spline, with the relationship $$ B^{k+1}_{i}(x)=…
Kane
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Euler equations in 2d

As an assignment in college, I did a 1d simulation. The problem statement was to solve 1d shock tube problem involving compressible ideal gas as working fluid. For this problem, I solved system of Eulers equations using Roe's Riemann solver. I want…
Subodh
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Solving a system with a small rank diagonal update

Suppose I have the original large, sparse linear system: $A\textbf{x}_0=\textbf{b}_0$. Now, I do not have $A^{-1}$ as A is too large to factor or any sort of decomposition of $A$, but assume that I have the solution $\textbf{x}_0$ found with an…
Costis
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3 answers

Methods of solving non-linear advection-diffusion systems beyond Newton-Raphson?

I'm working on a project where I have two adv-diff coupled domains through their respective source terms (one domain adds mass, the other subtracts mass). For brevity, I'm modeling them in steady state. The equations are your standard…
cbcoutinho
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What is a robust, iterative solver for large 3-d linear-elastic problems?

I'm diving into the fascinating world of finite element analysis and would like to solve a large thermo-mechanical problem (only thermal $\rightarrow$ mechanical, no feedback). For the mechanical problem, I already grasped from Geoff's answer, that…
Sebastian
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Parallel Mersenne Twister for Monte Carlo

Recently, I came across a comment claiming that almost all researchers doing Monte Carlo methods are doing it wrong. It went on to elaborate that merely choosing different seeds for different instances of a PRNG such as the Mersenne Twister is not…
Jonas Greitemann
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Does PETSc ever make use of LAPACK libraries for sparse matrix math?

Does compiling PETSc with an external BLAS/LAPACK library significantly affect performance on sparse matrices, or does it only use those libraries for dense matrix math?
Dan
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