Boundary Value Problems In Unbounded Domains Research Articles

A priori inequalities for solutions of mixed boundary-value problems in unbounded domains

We prove some a priori inequalities for solutions of mixed boundary-value problems for a class of divergence form elliptic equations with discontinuous and unbounded coefficients in unbounded domains.

Recent developments in finite element methods for structural acoustics

The study of structural acoustics involves modeling acoustic radiation and scattering, primarily in exterior regions, coupled with elastic and structural wave propagation. This paper reviews recent progress in finite element analysis that renders computation a practical tool for solving problems of structural acoustics. The cost-effectiveness of finite element methods is composed of several ingredients. Boundary-value problems in unbounded domains are inappropriate for direct discretization. Employing DtN methodology yields an equivalent problem that is suitable for finite element analysis by posing impedance relations at an artificial exterior boundary. Vell-posedness of the resulting continuous formulations is discussed, leading to simple guidelines for practical implementation and verifying that DtN boundary conditions provide a suitable basis for computation. Approximation by Galerkin finite element methods results in spurious dispersion, degrading with reduced wave resolution. Accuracy is improved by Galerkin/least-squares and related technologies on the basis of detailed examinations of discrete errors in simplified settings, relaxing wave-resolution requirements. This methodology is applied to time-harmonic problems of acoustics and coupled problems of structural acoustics. Space-time finite methods based on time-discontinuous Galerkin/least-squares are derived for transient problems of structural acoustics. Numerical results validate the superior performance of Galerkin/least-squares finite elements for problems of structural acoustics. A comparative study of the cost of computation demonstrates that Galerkon/least-squares finite element methods are economically competitive with boundary element methods, the prevailing numerical approach to exterior problems of acoustics. Efficient iterative methods are derived for solving the large-scale matrix problems that arise in structural acoustics computation of realistics configuration at high wavenumbers. An a posteriori error estimator and adaptive strategy are developed for time-harmonic acoustic problems and the role of adaptivity in reducing the cost of computation is addressed.

Nonlinear elliptic boundary-value problems in unbounded domains

Nonlinear elliptic boundary-value problems in unbounded domains

Uniqueness of the solutions of the cauchy problem and of boundary-value problems in unbounded domains for certain classes of nonlinear degenerate parabolic equations

The Cauchy problem and boundary value problems in unbounded domains are considered for certain classes of nonlinear degenerate parabolic equations, which contain as a particular case the equations of nonstationary filtration.