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### FEM-Laplace with Dirichlet in only a few points: Nonsingular operator?

Let's consider the FEM discretization of the Laplace operator without boundary conditions, i.e., $$a(u,v) = \int_\Omega \nabla u \cdot \nabla v - \int_\Gamma (n\cdot \nabla u) v.$$ For one-...
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67 views

### Estimating the local compression/expansion ratio for a transformation on a point cloud

Let's say we have an unorganized point cloud P1 with N points, each with coordinates {x,y,z}. We apply non-rigid transformation to P1 (translation + rotation + warping), to obtain point cloud P2. ...
658 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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### What is numerical damping in the context of time-dependent FEM solvers?

Comsol Multiphysics (a popular FEM package) includes two time-stepping algorithms (IDA aka BDF, and Generalized-alpha), described in their documentation as follows (quoted here under Fair Use; ...
807 views

### Which finite-element framework allows the easy implementation of HDG methods?

I have tinkered around with the implementation of hybridizable Discontinuous Galerkin (HDG) methods in DUNE-PDELab and DUNE-FEM for my PhD but since then I have not had the time to work on these codes ...
260 views

### Pseudo-inverse of a discretized operator with a null space?

Is there a way to understand what happens when a singular operator is discretized and inverted using the pseudoinverse (say using the SVD Moore-Penrose pseudoinverse)? For example, if we discretize ...
343 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 $0 < \varepsilon \ll 1$ (...
I have the following boundary value problem: $$-(\alpha u')' + \gamma u = f$$ in $\Omega = (a,b)$ with b.c. $u(a) = u(b) = 0$ and $\alpha > 0, \gamma ≥ 0$ and $f:(a,b) \to \Re$ The weak ...