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Accepted

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

Here's the deal with GPUs. On a GPU, every single core is slow. Really slow. However, you have thousands of cores. If you can effectively use the thousands of cores at a time, then your algorithm will ...
• 12.3k
Accepted

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

In the finite element method, the matrix entries and right hand side entries are defined as integrals. We can, in general, not compute these exactly and apply quadrature. But there are many quadrature ...
• 55.7k
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• 6,109
Accepted

### FEM for vector valued problems: reference request

Short answer: Just replicate the vector of interpolation functions into a block-diagonal matrix, as showed e.g. on page 5 in this lecture note. Detailed answer: Mathematically oriented texts typically ...
• 822

### Is it really necessary to solve a system of linear equations in the Finite Element Method?

I think your question is actually pretty fundamental and deserves a thoughtful answer. Paraphrasing a bit, your question is perhaps motivated by the observation that engineering design is often ...
• 4,906
Accepted

### Does the weighted residual method not use energy minimization in any form?

The answer to your question depends on the problem. For example, consider the diffusion equation for a field $q$, with diffusivity $k$ and sources $f$. The variational form of this problem states that,...
• 10.3k

### Why the product of symmetric-sparse matrices is not symmetric, or dense

You seem to think that: The product of two sparse matrices is sparse; The inverse of a sparse matrix is sparse; The product of two symmetric matrices is symmetric. None of these facts is true, in ...
• 11.5k
Accepted

### Solving Poisson equation with current BC using FEM

It is standard procedure for general case like that. You can enforce condition for electrode by Lagrange multipliers,  L(u,\lambda) = \int_\Omega \sigma u_{,i} u_{,i} \, \textrm{d}V - \int_{\...
• 906
Continuous finite elements Typically, if $A$ is your finite element discretization on the finest mesh, $A_i = R_i * A * R_i^T$. So, for $i=0$, $A_0$ corresponds to the finite element discretization ...