ageralo said:Homework Statement
Use FEM to solve this problem. The difficulty lies in the fact that the exact solution has a discontinuity in it. From x=[0,0.6) the exact solution u is x5/20 - x/20 and from x =(0.6,1] u is sin(x). The problem I'm having is I'm not sure what to do at the jump in my code. I have already set up my matrices Au = f, and it solves it fine given no discontinuity.
uxx = f
Homework Equations
u(0) = u0
u(1) = u1
FEM (Finite Element Method) with discontinuity is a numerical technique used to solve problems with discontinuities in the material properties, geometry, or loading conditions. It is often used in engineering and scientific fields to analyze structures and materials with complex and irregular features.
In FEM with discontinuity, the problem domain is divided into smaller elements and the properties and governing equations are defined for each element. At the interfaces of discontinuities, special techniques are used to connect the elements and ensure continuity of solutions. The governing equations are then solved numerically to obtain a solution for the entire problem domain.
FEM with discontinuity allows for more accurate and efficient analysis of complex problems compared to traditional analytical methods. It also allows for the inclusion of discontinuities, which are often present in real-world structures and materials.
FEM with discontinuity is commonly used in structural analysis, such as analyzing stress concentrations in joints or cracks in materials. It is also used in geotechnical engineering to study soil-structure interaction and in fluid mechanics to analyze flows with discontinuities.
One challenge of using FEM with discontinuity is the need for specialized techniques to handle the discontinuities, which may require additional computational resources and expertise. Another challenge is ensuring the accuracy and convergence of the solution, as some discontinuities may cause numerical instabilities in the solution process.