Nonlinear elliptic and evolution problems and their finite element approximations.

*(English)*Zbl 0731.65090
Computational Mathematics and Applications. London etc.: Academic Press, Inc. xix, 422 p. £45.00; $ 94.50 (1990).

The finite element analysis is performed for nonlinear elliptic and parabolic equations with strongly monotone elliptic operators and piecewise continuous coefficients in 2D bounded domains with Lipschitz continuous and piecewise smooth boundaries. The boundary conditions include nonhomogeneous mixed Dirichlet-Neumann conditions, the initial conditions are prescribed only in \(L^ 2(\Omega)\). In this fairly general frame the analysis includes besides the proofs of existence and uniqueness for each problem the following two basic steps:

(i) the proof of convergence for the approximate solutions under assumptions sufficient for existence and uniqueness and

(ii) error estimates under various regularity assumptions.

The course of investigations starts with two in some sense preliminary chapters dealing with the finite element method for linear elliptic problems. Particular attention is devoted to the study of triangular elements including constructive and analytic results as well as computational aspects. The analysis of elements consisting of a polynomial of first degree on triangles is prepared in chapter 3 and applied to the above mentioned classes of nonlinear stationary and evolutionary equations. There are some important and in other text books unavailable aspects:

(i) Systematic study of problems with piecewise continuous coefficients under theoretical and computational view,

(ii) Error estimates for nonlinear elliptic problems if the solution in \(H^ 1\) belongs piecewise to \(H^{1+\epsilon}\) and the corresponding subdomains are not polygonal, convergence proof for such problems using only that the solution is in \(H^ 1\) but allowing nonpolygonal domains, discontinuous coefficients and nonhomogeneous boundary conditions,

(iii) Maximum angle condition for triangular elements of Hermite type.

This is a fairly comprehensive, complete and clearly written work on the topic. Besides the careful mathematical analysis of the conforming finite element approximation the author presents in various connections a detailed explanation of the computational techniques.

(i) the proof of convergence for the approximate solutions under assumptions sufficient for existence and uniqueness and

(ii) error estimates under various regularity assumptions.

The course of investigations starts with two in some sense preliminary chapters dealing with the finite element method for linear elliptic problems. Particular attention is devoted to the study of triangular elements including constructive and analytic results as well as computational aspects. The analysis of elements consisting of a polynomial of first degree on triangles is prepared in chapter 3 and applied to the above mentioned classes of nonlinear stationary and evolutionary equations. There are some important and in other text books unavailable aspects:

(i) Systematic study of problems with piecewise continuous coefficients under theoretical and computational view,

(ii) Error estimates for nonlinear elliptic problems if the solution in \(H^ 1\) belongs piecewise to \(H^{1+\epsilon}\) and the corresponding subdomains are not polygonal, convergence proof for such problems using only that the solution is in \(H^ 1\) but allowing nonpolygonal domains, discontinuous coefficients and nonhomogeneous boundary conditions,

(iii) Maximum angle condition for triangular elements of Hermite type.

This is a fairly comprehensive, complete and clearly written work on the topic. Besides the careful mathematical analysis of the conforming finite element approximation the author presents in various connections a detailed explanation of the computational techniques.

Reviewer: H.Jeggle (Berlin)

##### MSC:

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

65N15 | Error bounds for boundary value problems involving PDEs |

35J65 | Nonlinear boundary value problems for linear elliptic equations |

35K55 | Nonlinear parabolic equations |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |

65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |