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## Regularity Assumptions (regularity + assumption)
## Selected Abstracts## Least-squares mixed finite element methods for the RLW equations NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 3 2008Haiming GuAbstract A least-squares mixed finite element (LSMFE) schemes are formulated to solve the 1D regularized long wave (RLW) equations and the convergence is discussed. The L2 error estimates of LSMFE methods for RLW equations under the standard regularity assumption on the finite element partition are given.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008 [source] ## New stabilized finite element method for time-dependent incompressible flow problems INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 2 2010*Article first published online: 20 FEB 200, Yueqiang ShangAbstract A new stabilized finite element method is considered for the time-dependent Stokes problem, based on the lowest-order P1,P0 and Q1,P0 elements that do not satisfy the discrete inf,sup condition. The new stabilized method is characterized by the features that it does not require approximation of the pressure derivatives, specification of mesh-dependent parameters and edge-based data structures, always leads to symmetric linear systems and hence can be applied to existing codes with a little additional effort. The stability of the method is derived under some regularity assumptions. Error estimates for the approximate velocity and pressure are obtained by applying the technique of the Galerkin finite element method. Some numerical results are also given, which show that the new stabilized method is highly efficient for the time-dependent Stokes problem. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## hp -Mortar boundary element method for two-body contact problems with friction MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 17 2008Alexey ChernovAbstract We construct a novel hp -mortar boundary element method for two-body frictional contact problems for nonmatched discretizations. The contact constraints are imposed in the weak sense on the discrete set of Gauss,Lobatto points using the hp -mortar projection operator. The problem is reformulated as a variational inequality with the Steklov,Poincaré operator over a convex cone of admissible solutions. We prove an a priori error estimate for the corresponding Galerkin solution in the energy norm. Due to the nonconformity of our approach, the Galerkin error is decomposed into the approximation error and the consistency error. Finally, we show that the Galerkin solution converges to the exact solution as ,,((h/p)1/4) in the energy norm for quasiuniform discretizations under mild regularity assumptions. We solve the Galerkin problem with a Dirichlet-to-Neumann algorithm. The original two-body formulation is rewritten as a one-body contact subproblem with friction and a one-body Neumann subproblem. Then the original two-body frictional contact problem is solved with a fixed point iteration. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## Non-linear initial boundary value problems of hyperbolic,parabolic type. MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 3 2002A general investigation of admissible couplings between systems of higher order.This is the second part of an article that is devoted to the theory of non-linear initial boundary value problems. We consider coupled systems where each system is of higher order and of hyperbolic or parabolic type. Our goal is to characterize systematically all admissible couplings between systems of higher order and different type. By an admissible coupling we mean a condition that guarantees the existence, uniqueness and regularity of solutions to the respective initial boundary value problem. In part 1, we develop the underlying theory of linear hyperbolic and parabolic initial boundary value problems. Testing the PDEs with suitable functions we obtain a priori estimates for the respective solutions. In particular, we make use of the regularity theory for linear elliptic boundary value problems that was previously developed by the author. In part 2 at hand, we prove the local in time existence, uniqueness and regularity of solutions to the quasilinear initial boundary value problem (3.4) using the so-called energy method. In the above sense the regularity assumptions (A6) and (A7) about the coefficients and right-hand sides define the admissible couplings. In part 3, we extend the results of part 2 to non-linear initial boundary value problems. In particular, the assumptions about the respective parameters correspond to the previous regularity assumptions and hence define the admissible couplings now. Moreover, we exploit the assumptions about the respective parameters for the case of two coupled systems. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## Non-linear initial boundary value problems of hyperbolic,parabolic type. MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 3 2002A general investigation of admissible couplings between systems of higher order.This is the third part of an article that is devoted to the theory of non-linear initial boundary value problems. We consider coupled systems where each system is of higher order and of hyperbolic or parabolic type. Our goal is to characterize systematically all admissible couplings between systems of higher order and different type. By an admissible coupling we mean a condition that guarantees the existence, uniqueness and regularity of solutions to the respective initial boundary value problem. In part 1, we develop the underlying theory of linear hyperbolic and parabolic initial boundary value problems. Testing the PDEs with suitable functions we obtain a priori estimates for the respective solutions. In particular, we make use of the regularity theory for linear elliptic boundary value problems that was previously developed by the author. In part 2, we prove the local in time existence, uniqueness and regularity of solutions to quasilinear initial boundary value problems using the so-called energy method. In the above sense the regularity assumptions about the coefficients and right-hand sides define the admissible couplings. In part 3 at hand, we extend the results of part 2 to the nonlinear initial boundary value problem (4.2). In particular, assumptions (B8) and (B9) about the respective parameters correspond to the previous regularity assumptions and hence define the admissible couplings now. Moreover, we exploit assumptions (B8) and (B9) for the case of two coupled systems. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## Stochastic homogenization of Hamilton-Jacobi-Bellman equations COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 10 2006Elena KosyginaWe study the homogenization of some Hamilton-Jacobi-Bellman equations with a vanishing second-order term in a stationary ergodic random medium under the hyperbolic scaling of time and space. Imposing certain convexity, growth, and regularity assumptions on the Hamiltonian, we show the locally uniform convergence of solutions of such equations to the solution of a deterministic "effective" first-order Hamilton-Jacobi equation. The effective Hamiltonian is obtained from the original stochastic Hamiltonian by a minimax formula. Our homogenization results have a large-deviations interpretation for a diffusion in a random environment. © 2005 Wiley Periodicals, Inc. [source] |