A Posteriori Error Estimation and Corrected Solution

APIC'95, El Paso, Extended Abstracts, A Supplement to the international journal of Reliable Computing

A Posteriori Error Estimation and Corrected Solution of Partial Dierential Equation Boris S. Dobronets

II. A Numerical Solution Of The Abstract| In this paper, we show that Boundary Value Problem a posteriori error estimate technique can be used to produce corrected solutions of To describe the solution, we must rst de ne the boundary value problem for partial dieren- nite element space S . We x a mesh size h, and tial equations. divide the domain into subdomains of size h. The set of these subdomains will be denoted by T . The n l

I. Formulation Of The Boundary Value Problem

We will illustrate our idea on the following elliptic boundary value problem Lu = f (x; u); x 2 ; u(x) = 0; x 2 @ ;

(1) (2)

where is a bounded open convex domain in R2, with piecewise smooth boundary @ , and Lu =

X @ u=@x : 2

i

2

2 i

=1

h

nite-element space is de ned as a set of piece-wise polynomial functions (i.e., function that are polynomial on each of the subdomains T 2 T ): S = f s(x) j s 2 W2 ( ); sj 2 P ; T 2 T g; (4) where P is the set of polynomials of degree n. We de ne the nite element solution u of the problem (1) as a function from S11 that satis es the conditions L(u ; v ) = (f (x; u (x)); v ); 8v in S11 ; (5) u = 0; on @ : Since the function u is uniquely determined by its ( nitely many) polynomial coecients, we have a (non-linear) system of equations, that can be solved, e.g., if we use iterative techniques. h

n l

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n

T

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h

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h

h

h

In this paper, we assume that @f (x; u)=@u  q(x)  0;

and that a positive constant K exists such that

III. Estimating Error Of This Numerical Solution

We would like to use the defect Lu f (x; u ) for all x 2 , and for all  2 [min u; max u]: The solution of (1) is understood in the following of the approximate solution u to estimate its accuracy, i.e., the dierence between this approximate weak sense: nd a function u 2 W21 ( ) such that solution and the (unknown) precise one. However, L(u; v) = (f; v); 8v 2 W21 ( ); (3) we cannot compute the defect of the original solution, because by de nition, the function u is dierentiable only once, so, we cannot compute its second where (
;
) is the inner product in L2: derivatives that form an operator L. 2 So, to apply this idea, we use the nite element L(u; v) = @ u @ v d : solution u to construct a new smooth approximate

=1 solution s 2 S23 ( ). Informally speaking, this solution s must satisfy two conditions: The author is with Computer Center of Siberian Department of Russian Academy of Sciences, Akademgorodok, 660036, Kras First, it must be close to the original solution noyarsk, Russia, E-mail: [email protected]. This work u ; i.e., for every mesh point v , we must have was partially supported by Krasnoyarsk Regional Science Foundation, Krasnoyarsk, Russia. s(v )  u (v ).

jf (x; )j  K (1 + jj);

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APIC'95, El Paso, Extended Abstracts, A Supplement to the international journal of Reliable Computing

 Second, this function s must be an (approx- where L1 = Lu qu. We solve this auxiliary problem

imate) solution of the equations (1). This means, in particular, that for every mesh point v , we must have Ls(v )  f (x; s(v )). The function f is non-linear, so, this condition is non-linear in s. However, we can linearize it if we take into consideration that s(v )  u (v ). Then, this condition can be reformulated as follows: Ls(v )  f (v ; u (v )). We cannot exactly satisfy both sets of conditions, so we must apply a techniques similar to least squares method to combine them. As a result, we arrive at the following method of determining s: For an arbitrary mesh point x0 = (x01; x02), we de ne the (local) polynomial solution p as a polynomial p= a (x1 x01) (x2 x02 ) i

i

X

i;j

such that

h

i

h

i

X  jp(v ) N

i

=1

M

i

i

=1

j

u (v )j2+ h

i

X jLp(w ) i

i

h

i

i

i

i

i

M i

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i

i

i

 = max(=(L1 s1 ); 0);  = min(=(L1s1 ); 0);

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@

;

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IV. How To Correct The Solution Of The Boundary Value Problem

To correct the solution, we must somehow nd a good approximation for the dierence u s between the exact (unknown) and the approximate solutions. To estimate this dierence, let us nd a dierential equation that this function " = u s must satisfy. If we substitute u = s + " into the equation (1), we get the following equation:

L" = f (x; s + ") Ls:

To estimate ", we can esither solve this non-linear equation (using the same techniques as above), or simplify it. We can simplify this non-linear equation if we take into consideration the fact that " is small and therefore, we can expand f into a power series in " and keep only linear terms. As a result, we get the following equation: L" = f (x; s) + r(x)" Ls;

ij

ij

;



= max( s1 s; 0); = min( s1 s; 0);



(6) or

where fv g =1 and fw g =1 are nodes of the mesh,  = a1 =((v ; x0) + a2 ); = b1 =((w ; x0) + b2 ); a and b are positive constants, and (x; y) denotes the distance between the points x; y. The problem (6) is quadratic in the unknown coecients a . Thus, if we dierentiate with respect to a , and equate the derivatives to 0, we thus reduce this problem to the solution of the following system of linear algebraic equations: Ba = d; where a = (a0 0; a1 0; a0 1; : : :; a 0; a 1 1; : : :; a0 ); d = (u1 ; : : :; u ; f (w1 ; u (w1)); : : : ; f (w ; u (w ))): After this smooth approximate solution s is computed, we can use the following defect [2] (x; s) = Ls f (x; s); x 2 : (7) To get the desired interval estimate, we apply the nite element method (FEM) to solve numerically the following additional problem: L1 u1 = 1; x in ; (8) u1(x) = 0; x on @ ; (9) N i

where

Ls + L" = f (x; s + ");

f (w ; u (w ))j2 ! min;

i

i

i

i;j

:0i+j n

u = s + [; ]s1 + [ ; ];

i

i

i

by constructing a special spline s1 . Then, the desired interval solution has the form

where we denoted r(x) =

@f (x; s(x)): @u

If we move all the terms that contain u into the left-hand side, and all the other terms into the righthand side, we get the following equation: L" r(x)" = (x);

M

where (x) = f (x; s(x)) (Ls)(x): Let " be a solution of this approximate equation, obtained by a similar nite-element technique. Then, for the corrected solution h

s

cor

= s+" ;

the following result is true:

h

APIC'95, El Paso, Extended Abstracts, A Supplement to the international journal of Reliable Computing

THEOREM. If u 2 W24( ), then ku s k  C
h4kuk 24 ( ); cor

W

where h is the mesh size, and C is a constant independent of h.

To get an even better solution, we can apply the correction procedure to the corrected solution; then, if necessary, we can apply the correction procedure again and again, and get better and better solutions. In principle, instead of starting with an approximate solution, we can start with the u = 0 and s = 0. Thus, we will get a new iterative method of numerical solution for such equations. h

References

[1] B. S. Dobronets, \Dierence correction by nite elements of higher order", In: Mathe-

matical models and methods of solving the problems of continuous media, Krasnoyarsk, 1986, pp. 80{88 (in Russian). [2] B. S. Dobronets, V. V. Shaydurov, Two-sided Numerical Methods, Nauka Publ. (Siberian Branch), Novosibirsk, 1990 (in Russian). [3] G. Streng and G. F. Fix, An Analysis of the Finite Element Method, Englewood Clis, Prentice-Hall, 1973.