Well-Posed Problem in Partial Differentrial Equations

To avoid the Ambiguity appearing in the Formulation Process using the Numerical Methods of Partial Differentrial Equations it is Important to Satisfy the following Conditions given by Fletcher C.A.J. 1989, p18 :

"The governing Equations and Auxiliary (Initial and Boundary) Conditions are Well-Posed Mathematically if the Following three Condtions are met:

• the solution exists,
• the solution is unique,
• the solution depends continuously on the auxiliary data

The above criteria are usually attributed to Hadamard[Garabedian P. 1964, p109]

There are some Flow Problems for which Multiple Solutions may be expected on Physical Grounds. These Problems would fail the above Criteria of Mathematical Well-Posedness. The Computation may be Complicated by concern about the Well-Posedness of the Mathematical Formulation

In addition we could take a simple parallel and require that for Well-Posed Computation :

• the computatinal solution exists,
• the computational solution is unique,

• the computational solution depends continuously on the approximate auxiliary data."

The Finite Difference Formulations Applied to Fluid Mechanics can be obviously Applied to Solid Mechanics.

Mohammed lamine Moussaoui