What is the term for the integral statement of a PDE multiplied by a weighting function and equated to zero?

What is the term for the integral statement of a PDE multiplied by a weighting function and equated to zero?

• A. Strong Form
• B. Variational Form
• C. Discretized Form
• D. Weak Form

Answer: (D)

Formulating a PDE in its weak form means turning the equation into an integral statement by multiplying the residual with a weighting (test) function and setting it to zero for all such test functions. This approach shifts the problem from satisfying the differential equation pointwise everywhere to satisfying it in an averaged sense over a chosen function space. The weighting function serves to probe the residual, and requiring the integral to vanish for every allowable test function enforces a solution that is compatible with the underlying physics, even if the solution isn’t smooth enough to satisfy the strong form. For example, consider a Poisson problem. Multiplying by a test function and integrating, then using integration by parts, yields an equality that must hold for all test functions. This is the weak form: a bilinear form involving the gradients of the unknown and test function equals a linear form of the test function. It’s the framework finite element methods use, because it accommodates less regular solutions and leads to a solvable system after discretization. Sometimes people call this the variational form, since it arises from a variational principle, but the phrase that specifically describes the integral, weighted-residual statement equal to zero is the weak form. The strong form would be the original differential equation, and the discretized form refers to the finite-dimensional algebraic system that results once you approximate the function spaces.

Formulating a PDE in its weak form means turning the equation into an integral statement by multiplying the residual with a weighting (test) function and setting it to zero for all such test functions. This approach shifts the problem from satisfying the differential equation pointwise everywhere to satisfying it in an averaged sense over a chosen function space. The weighting function serves to probe the residual, and requiring the integral to vanish for every allowable test function enforces a solution that is compatible with the underlying physics, even if the solution isn’t smooth enough to satisfy the strong form.

For example, consider a Poisson problem. Multiplying by a test function and integrating, then using integration by parts, yields an equality that must hold for all test functions. This is the weak form: a bilinear form involving the gradients of the unknown and test function equals a linear form of the test function. It’s the framework finite element methods use, because it accommodates less regular solutions and leads to a solvable system after discretization.

Sometimes people call this the variational form, since it arises from a variational principle, but the phrase that specifically describes the integral, weighted-residual statement equal to zero is the weak form. The strong form would be the original differential equation, and the discretized form refers to the finite-dimensional algebraic system that results once you approximate the function spaces.