# Weak Form of the Poisson Problem

• Morberticus
In summary, the weak form of the slightly more complicated Poisson problem is given by the product rule and is similar to the well-known form for the simpler Poisson problem. This form is useful in writing finite element code and considering variational/weak forms.
Morberticus
Hi,

I know the weak form of the Poisson problem

$\nabla^2 \phi = -f$

looks like

$\int \nabla \phi \cdot \nabla v = \int f v$

for all suitable $v$. Is there a similarly well-known form for the slightly more complicated poisson problem?

$\nabla (\psi \nabla \phi ) = -f$

I am writing some finite element code and variational/weak forms are very handy.

Last edited:
Morberticus said:
Hi,

I know the weak form of the Poisson problem

$\nabla^2 \phi = -f$

looks like

$\int \nabla \phi \cdot \nabla v = \int f v$

for all suitable $v$. Is there a similarly well-known form for the slightly more complicated poisson problem?

$\nabla (\psi \nabla \phi ) = -f$

I am writing some finite element code and variational/weak forms are very handy.

By the product rule
$$\int_V v\nabla \cdot(\psi \nabla \phi)\,dV = \int_V\nabla\cdot(v\psi \nabla \phi) - \psi (\nabla \phi) \cdot (\nabla v)\,dV = \int_{\partial V} v\psi \nabla \phi \cdot dS - \int_V\psi (\nabla \phi) \cdot (\nabla v)\,dV$$
and hence the weak form of $\nabla \cdot(\psi\nabla\phi) = - f$ is
$$\int_V \psi (\nabla \phi) \cdot (\nabla v)\,dV = \int_V fv\,dV$$

1 person

## 1. What is the weak form of the Poisson problem?

The weak form of the Poisson problem is a mathematical technique used to solve for the solution of the Poisson equation, which is a second-order partial differential equation commonly used in physics and engineering. It involves transforming the original problem into a variational problem, where the solution is found by minimizing an associated functional.

## 2. How is the weak form different from the strong form of the Poisson problem?

The strong form of the Poisson problem is the original differential equation that must be satisfied for a solution to exist. The weak form, on the other hand, is a reformulation of the strong form that allows for a more general solution by introducing a test function and relaxing the requirement of the solution to satisfy the differential equation at every point in the domain.

## 3. What are the benefits of using the weak form of the Poisson problem?

One benefit of using the weak form is that it allows for a wider range of solutions to be considered, as it relaxes the strict requirements of the strong form. Additionally, the weak form can be solved using a variety of numerical methods, making it a useful tool for solving complex problems in physics and engineering.

## 4. What are some applications of the weak form of the Poisson problem?

The weak form of the Poisson problem has many applications in physics and engineering, including solving problems in fluid dynamics, heat transfer, and structural analysis. It is also commonly used in finite element analysis, a numerical method for solving partial differential equations.

## 5. What are some limitations of the weak form of the Poisson problem?

One limitation of the weak form is that it may produce a less accurate solution compared to the strong form, as it allows for a wider range of potential solutions. Additionally, it may be more computationally expensive to solve compared to the strong form. However, these limitations can often be mitigated by using advanced numerical methods and techniques.

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