ISSN 0012-2661, Differential Equations, 2017, Vol. 53, No. 7, pp. 964–973.
Pleiades Publishing, Ltd., 2017.
Original Russian Text
P.P. Matus, D.B. Poliakov, 2017, published in Differentsial’nye Uravneniya, 2017, Vol. 53, No. 7, pp. 991–1000.
Consistent Two-Sided Estimates
for the Solutions of Quasilinear Parabolic Equations
and Their Approximations
P. P. Matus
Institute of Mathematics, National Academy of Sciences of Belarus, Minsk, 220072 Belarus
The John Paul II Catholic University of Lublin, Lublin, 20-950 Poland
Received January 24, 2017
Abstract— For a linearized ﬁnite-diﬀerence scheme approximating the Dirichlet problem for
a multidimensional quasilinear parabolic equation with unbounded nonlinearity, we establish
pointwise two-sided solution estimates consistent with similar estimates for the diﬀerential prob-
lem. These estimates are used to prove the convergence of ﬁnite-diﬀerence schemes in the grid
The maximum principle allows one not only to establish the uniqueness of the solution and its
continuous dependence on the input data for parabolic and elliptic equations but also, in contrast
to the energy inequality method, to obtain a priori upper bounds for the solution in the uniform
norm for such problems of arbitrary dimension with a nonself-adjoint elliptic operator [1, p. 500].
The maximum principle is also used in the theory of ﬁnite-diﬀerence schemes to study the stability
of the ﬁnite-diﬀerence solution with respect to the input data and its convergence to the exact
solution of the problem in the uniform norm. Finite-diﬀerence methods satisfying the grid maximum
principle are usually said to be monotone [2, p. 228; 3, p. 296]. Various classes of monotone
ﬁnite-diﬀerence schemes have been developed and studied for multidimensional linear convection–
diﬀusion equations (e.g., see the monograph [4, p. 35]). Monotone schemes play an important role
in computational practice in that they allow one to obtain oscillation-free numerical solutions even
in the case of nonsmooth solutions .
Lower (or, in the general case, two-sided) estimates of solutions of diﬀerential-diﬀerence problems
are of no less importance. Such estimates are especially important when studying the properties
of numerical methods for problems with unbounded nonlinearity, because in this case one needs
to establish that the grid solution lies in a neighborhood of the values of the exact solution [6, 7].
For linear problems, these estimates enable one to ﬁnd the range of values of the desired solution
in terms of the problem input data (the coeﬃcients and right-hand side of the equation as well as
the initial and boundary conditions). In the nonlinear case, such estimates permit one to prove the
nonnegativity of the exact solution, which is important in physical problems, as well as to ﬁnd
conditions on the input data under which the problem is parabolic or elliptic.
Finding nontrivial estimates for the solutions of initial–boundary value problems is based on
a special trick, originally applied by Ladyzhenskaya  (see also the monograph [9, p. 22]), whereby
one makes a parameter-dependent change of variables and then minimizes or maximizes some func-
tions with respect to this parameter; the resulting extremal values give the corresponding estimates
for the solution. Naturally, one also needs such estimates in numerical algorithms for the approxi-
mate solution of initial–boundary value problems. The theory of ﬁnite-diﬀerence schemes [2, p. 229]
includes the technique, well developed for linear problems, of the grid maximum principle, which