# Finite Element Method Questions and Answers – Weak Formulation of Boundary Value Problems

This set of Finite Element Method Multiple Choice Questions & Answers (MCQs) focuses on “Weak Formulation of Boundary Value Problems”.

1. What is considered as the first step in weak formulation of boundary conditions?
a) Multiplication by arbitrary function
b) Integration by parts
c) Integration over the entire domain
d) Application of boundary conditions

Explanation: In weak formulation of boundary conditions, the partial differential equation is first multiplied with an arbitrary function. Following this it undergoes integration over the whole domain supported by integration with parts. The boundary conditions are applied just before solving the problem.

2. Why is integration by parts method preferred in weak formulation?
a) It helps formulate asymmetric coefficient stiffness matrices
b) It helps formulate symmetric coefficient stiffness matrices
c) It helps formulate symmetric coefficient mass matrices
d) It helps formulate asymmetric coefficient mass matrices

Explanation: By making use of integration by parts method, we succeed in having the same order for both the partial differential equations under consideration. When the equations have similar orders, it leads to formation of symmetric coefficient stiffness matrices. This is favored while solving finite element problems.

3. In certain scenarios, weak formulation is preferred. Why is this so?
a) It poses high continuity requirements
b) It doesn’t have any requirements
c) It poses lower continuity requirements
d) It has minimal power requirements

Explanation: The weak form poses lower continuity requirements for the chosen approximate solution. This can be proven by taking note of the single order derivatives present in the weak form; whereas the strong form possesses second order derivatives which require the approximate function to be continuous.

a) Functions that satisfy strong formulation requirements
b) Functions that do not satisfy weak formulation requirements
c) Functions that do not satisfy strong formulation requirements
d) Functions that satisfy weak formulation requirements

Explanation: Functions that satisfy weak formulation requirements are referred to as admissible functions. However, theoretically out of the two functions [u(x) and w(x)], w(x) is not expected to be admissible. u(x) is expected to exist at the essential boundary conditions, while w(x) is expected to vanish at the essential boundary.

5. In finite element approximation of weak form, what are the two functions referred to as?
a) u(x) = trial solution and w(x) = test function
b) u(x) = test function and w(x) = trial solution
c) u(x) = w(x) = weight functions
d) u(x) = w(x) = residual functions

Explanation: When finite element approximation of weak form is taking place, each function is assigned a new term/name. The u(x) corresponds to the trial solution; whereas, the w(x) corresponds to the test function of the weakly formulated equation under discussion.

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