Problem 4 - Concepts and Applications of Finite Element Analysis

Question

Consider the differential equation \(u_{x, x}+4 u=12\), with essential boundary conditions \(u=3\) at \(x=0\) and \(u=1\) at \(x=1\). There are no nonessential boundary conditions. The exact solution is \(u=3-2.1995 \sin 2 x .\) A oneparameter approximating polynomial that meets the essential boundary conditions is \(\bar{u}=3-2 x+a\left(x^{2}-x\right)\). Determine parameter \(a\) in the range \(0

Step-by-Step Solution

Answer

The solution to this exercise requires the application of five different methods of approximation to the given differential equation, followed by calculation of the percentage error for each method at specified points. The details of each method's application and error calculation would depend on the values obtained at each step.

Step 1: Collocation

In the collocation method, choose the point \(x=0.5\) and ensure the differential equation is satisfied at this point. By substituting \(u_{xx} = -4u + 12\) and \(u = 3 - 2x + a(x^2 - x)\) at \(x=0.5\) in the main differential equation, and solving for \(a\), then use the exact solution to calculate the percentage error at \(x=0.5\) and \(x=0.7\).

Step 2: Subdomain

In the subdomain method, take the mean value of the function and its approximation over the interval [0,1] and set them equal. By taking the mean value of the left hand side and right hand side of the main differential equation, we get two equations in two variables: one for the mean value of \(u\) and one for the mean value of \(u_{xx}\). Solve these two equations simultaneously to get the value of \(a\), then use the exact solution to calculate the percentage error at \(x=0.5\) and \(x=0.7\).

Step 3: Least squares

In this method, we define a residual \(R=a(x^2-x)-u\) where \(u\) satisfies the given differential equation and then find the value of \(a\) that minimizes the integral of \(R^2\) over the interval [0,1]. This minimization problem can be solved by setting the derivative of the integral with respect to \(a\) equal to zero, and solving for \(a\), then use the exact solution to calculate the percentage error at \(x=0.5\) and \(x=0.7\).

Step 4: Least squares collocation

Choose points \(x=\frac{1}{3}\) and \(x=\frac{2}{3}\) and ensure the residual \(R=a(x^2-x)-u\) is a minimum at these points. Solve the resulting system of equations for \(a\), then use the exact solution to calculate the percentage error at \(x=0.5\) and \(x=0.7\).

Step 5: Galerkin method

In this method, the given equation is multiplied by a weight function and then integrated over the domain [0,1]. The resulting equation is then solved for \(a\). For this problem, the weight function is chosen to be \(w = x^2 - x\), the same as in the approximation function. After solving for \(a\), use the exact solution to calculate the percentage error at \(x=0.5\) and \(x=0.7\).

Key Concepts

Boundary Conditions

Boundary conditions are an essential part of solving differential equations as they provide the necessary information to find a unique solution. In this exercise, we have essential boundary conditions specified as \( u=3 \) at \( x=0 \) and \( u=1 \) at \( x=1 \). These conditions tell us the values that the function \( u \) must take at the boundaries of the domain.

There are two types of boundary conditions: essential (or Dirichlet) and non-essential (or Neumann). Essential boundary conditions specify the value of the function at certain points, while non-essential conditions specify the value of the derivative of the function.

• In our exercise, only essential boundary conditions are present.
• No non-essential boundary conditions are specified, implying that the solution only requires the values of \( u \) at the boundaries.

Understanding and properly applying boundary conditions are crucial as they ensure that the solution aligns with the physical constraints and requirements of the problem.

Collocation Method

The collocation method is a technique used for approximating the solutions of differential equations. This method involves selecting specific points, called collocation points, where the approximate solution must satisfy the given equation.

In this problem, the point \( x=0.5 \) is used as the collocation point. At this point:

• The differential equation \( u_{x,x} + 4u = 12 \) must be satisfied using the approximating polynomial \( \bar{u} = 3 - 2x + a(x^2 - x) \).
• By substituting these expressions into the differential equation at \( x=0.5 \), we solve for the parameter \( a \).

The collocation method simplifies the task by reducing a complex problem into a series of simpler ones at chosen discrete points. It is a widely used method in engineering applications for its simplicity and effectiveness.

Differential Equations

Differential equations are mathematical equations that involve the rates of change of quantities and the quantities themselves. The exercise deals with a second-order differential equation \( u_{x,x} + 4u = 12 \). This equation involves the second derivative of \( u \) with respect to \( x \), which represents the rate of change of the slope of \( u \).

Solutions to differential equations are critical in modeling and predicting real-world phenomena where changes over time or space are involved.

• Second-order differential equations, like the one in this problem, often arise in physics, engineering, and other sciences.
• To solve it, one often needs initial conditions or boundary conditions to find a unique solution.

In this exercise, we use approximating functions and methods like collocation and the Galerkin method to find approximate solutions for these complex equations.

Galerkin Method

The Galerkin method is an approach used to find approximate solutions to differential equations by using weighted residuals. In this method, the differential equation is multiplied by a weight function and integrated over the domain.

For this exercise:

• The weight function chosen is \( w = x^2 - x \), which matches the form of the approximating polynomial.
• Multiply the entire differential equation by this weight function and integrate over the interval \([0,1]\).
• This integration results in an equation that allows solving for the parameter \( a \).

By choosing the weight function equal to the basis function of the approximation, we ensure that the error in the solution is minimized in an average sense across the entire domain. This makes the Galerkin method particularly effective in complex real-world applications where precision is required over broad areas or intervals.