Questions tagged [mixed-formulation]

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positivity preservation for mixed finite element discretization

I'm interested in mixed discretizations of the diffusion (and related) equations: $$\begin{align} \frac{\partial h}{\partial t} + \nabla\cdot \mathbf q & = f \\ k^{-1}\mathbf q + \nabla h & = ...
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1 vote
2 answers
88 views

Taylor-Hood elements for Darcy's equation

I would like to know if Taylor-Hood elements $P_2$-$P_1$ form a stable pair for the mixed approximation of Darcy's equation ( or Poisson's equation) with Dirichlet B.C. In the literature I only find ...
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2 votes
1 answer
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Distributed Lagrange multiplier approach to impose constraint in Poisson equation

I'm trying to understand how Lagrange multipliers are applied in order to impose constraints in PDEs. Consider $B \subset \Omega$. For instance, a square inside another square domain $\Omega$. Let's ...
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2 votes
1 answer
202 views

Solving Poisson equations as mixed Laplace using $RT_0-P_0$ pair

I'm trying to solve \begin{cases} - \Delta p=f \text{ in } \Omega\\ p=0 \text{ on } \partial \Omega \end{cases} with in $\Omega = [-1,1]^2$ by writing it as \begin{cases} u + \nabla p=0 \\ -\...
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6 votes
2 answers
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Is their a book/paper about mixed finite element method for engineering students (non-math)?

There are a lot of books about FEM, which are really friendly to engineering students. Through these books, we can know how to use shape/test functions based on the variational principle. But I'd like ...
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1 vote
1 answer
110 views

Locking phenomena for $P1 - P0$ elements

Consider the Stokes problem and the usual divergence operator $B:V \rightarrow Q'$, $\langle Bv, q\rangle = b(v,q)=(\operatorname{div} v,q)$ and its discrete versione $B_h : V_h \rightarrow Q_h'$. In ...
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5 votes
0 answers
147 views

About the condition $\ker(B_h) \subset \ker(B)$ in mixed finite elements formulation

I'm studying mixed finite elements. The problem is a classical saddle-point one: we seek for $(u,p)$ in $V \times Q$: $$A u + B^t p = f$$ $$Bu = g$$ where $A: V \rightarrow V', B:V \rightarrow Q'$ ...
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1 vote
3 answers
151 views

Stable finite elements for the mixed form of the elasticity equations

The mixed form of the elasticity equations is to find the unique critical point of the Hellinger-Reissner functional $$J(u, \sigma) = \int_\Omega\left(\frac{1}{2}A\sigma : \sigma - (\nabla\cdot\sigma)\...
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