# 4 interpolation error estimates interpolation error

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 0] describe many of their properties, similarities, and di erences. Let us set the stage by brie y describing two simple extrapolation techniques. Consider a p one-dimensional problem for simplicity and suppose that an approximate solution Uh (x) has been computed using a polynomial approximation of degree p on a mesh of spacing h (Figure 7.4.1). Suppose that we have an a priori interpolation error estimate of the form p u(x) ; Uh (x) = Cp+1hp+1 + O(hp+2): We have assumed that the exact solution u(x) is smooth enough for the error to be expanded in h to O(hp+2). The leading error constant Cp+1 generally depends on (unknown) derivatives of u. Now, compute a second solution with spacing h=2 (Figure 7.4.1) to obtain p u(x) ; Uh=2 (x) = Cp+1( h )p+1 + O(hp+2): 2 18 Analysis of the Finite Element Method U2 h U1 h/2 U1 h h x 1 1 Figure 7.4.1: Solutions Uh and Uh=2 computed on meshes having spacing h and h=2 with 2 piecewise linear polynomials (p = 1) and a third solution Uh computed on a mesh of spacing h with a piecewise quadratic polynomial (p = 2). Subtracti...
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern