Numerical Analysis on Non-Smooth Problems: Some Examples
a rough right hand side,
a rough boundary,
rough initial data. We shall briefly describe two examples from numerical analysis showing that:
Example A) In a linear Poisson problem, roughness introduced by i) or ii) leads to dramatically different behavior in numerical solutions.
Example B) Although for linear and semilinear parabolic problems, in the case of iii) (all other “data” being smooth) there is “no difference” in the sense that the solution is smooth for (some) positive time, in numerical analysis the linear and semilinear problem are miles apart.
While the main point of Example A was “computational folklore” for many years before it was fully proven in 1984, the punchline of Example B was totally unexpected at the time of its discovery in 1987.
More details, and more examples of rough stuff in the numerical analysis of partial differential equations, e.g., in singularly perturbed problems, can be found in WAHLBIN .
KeywordsFinite Element Solution Positive Time Finite Element Space Duality Argument Essential Boundary Condition
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- Thomée, V., Spline approximation and difference schemes for the heat equation in: A.K. Aziz, ed., The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, Academic Press 1972, 711–746.Google Scholar
- Wahlbin, L.B., A brief survey of parabolic smoothing and how it affects a numerical solution: finite differences and finite elements, in: I. Babuška, T.-P. Liu and J. Osborn, eds., Lectures on the Numerical Solution of Partial Differential Equations, University of Maryland Lecture Notes 20, 1981.Google Scholar
- Wahlbin, L.B., Local Behavior in Finite Element Methods, in: P.G. Ciarlet and J.-L. Lions, eds., Handbook of Numerical Analysis, Vol. II, Finite Element Methods, Elsevier (North Holland), to appear in 1990.Google Scholar