Numeric method to select an analytic polynomial high degree solution for Linear or Nonlinear Elliptic Partial Differential Equations

Maty Blumenfeld, Eng., Ph.D. is Professor Emeritus at the Department of Strength of Materials, University POLITEHNICA of Bucharest and one of the associates of INAS SA. He solved numerous technical and scientific problems and published many books dealing with topics primarily concerning the Strength of Materials, but also on the domains of Finite Element Method and Numerical Methods. His latest book, Numeric method to select an analytic polynomial high degree solution for?Linear or Nonlinear Elliptic Partial Differential Equations, was just released and can be downloaded from here: https://www.blumenfeld.ro/. It has such a long title for the following reasons:

1. The title emphasizes that a NUMERICAL method is described, but that the result of its use is an ANALYTICAL SOLUTION of an Elliptic Differential Equation (PDE), which is unusual. The analytical solution is obtained USING A SINGLE ELEMENT. The solution has the form of a complete POLYNOMIAL of very high degree, which makes that it includes a very large number of terms. Unlike using a single element, the Finite Element Method (FEM) divides the integration domain in hundred thousands or millions of elements.

2. The degree of the polynomial is not obtained directly, but by SELECTING the best solution from 5 successive solutions of PDE, made with polynomials of degree D=6,7,8,9 or 10. The selected solution can comprise between 28 terms (corresponding to degree D=5 of the polynomial) and 66 terms (degree D=10). The computation of the function value at any point of the integration domain is done - as for any analytical function - by replcing in the solution the coordinates of the point. This, unlike FEM that gives discrete results, represented by millions of values.

?3. The method can solve second order LINEAR or NONLINEAR PDEs. A linear elliptic PDE contains the following 7 terms

??????????????????????(1)

The coefficients of the first 6 terms that depend on the unknown function Φ(x,y), noted (a,b,c,M,N,P), are considered as constants, while the last term W(x,y) is a known two-dimensional function. The PDE (1) is ELLIPTIC if??

RR =b2-4ac ?< 0????????????????????????????????????????????????????????(2)

The following second order elliptic equation that has to be integrated on a square domain?1×1

?????????????????????????(3)

has as ?selected ?solution ?a ?10 degree polynomial. The solution can be visualized if it is represented graphically, like in the following drawing

The choice of the degree of the polynomial is based on the appreciation of the precision of the solution. Although the method used is NUMERICAL, the selected solution being a function, the determination of the precision of the result is done as in the ANALYTICAL methods, ie by replacing the solution in the initial equation. Suppose that the computed solution selected for PDE (1) coincide – by chance –with the exact function Φ(x,y). In this case, if the solution is replaced in (1) the result ?will be, obviously, PDE=0.?If?it is found by the computation a nonidentical but close solution z≈ Φ, the resulted value, noted as Residual, will be different from zero

???????(4)

The use of the Residual presents the advantage that it can take into account not only the function z(x,y) selected by the computation, but?also the first and second partial derivatives of it.

Because the selected solution is always a polynomial of high degree, a SAFE method has been created to assess the efficient use of the Residual: a PDE is composed, having as known solution a NONPOLYNOMIAL function.

Because the selected solution is always a high-grade polynomial, a safe method has been created to evaluate the efficient use of the Residual: a PDE is composed, having as known solution a NONPOLINOMIAL function.

The use of the Residual allows a new approach to the problem of?accuracy of the integration result. Inspired by the method of allowable resistance used in Strength of Materials, the author proposes an admissible value for the Residual, which separates the good results from those considered as inappropriate. Obviously, this admissible value is disputable and will be – probably - modified, when other users will express their point of vue. For the widest possible spread of the method, the author sought to present in detail the main relations, which would allow any graduate of a mathematics or engineering faculty to compile his own program.

The NONLINEAR PDE

???????????????????(5)

has to be integrated on a domain 5×5. The solving methodology starts by transforming the PDE into a linear equation

????(6)

whose solution is represented in Fig.A. Starting from this first attempt (that may be far from the truth), an iterative procedure is applied, which if it is convergent, leads - in a ?short time - to the solution of the nonlinear equation. The procedure being convergent in the case of PDE (5) the NONLINEAR answer is found after 11 iterations (Fig.B).

Fig.A. Linear PDE (6)?????????????????????

Fig.B. Nonlinear PDE (5)

Complete book can be downloaded from: https://www.blumenfeld.ro/