Paul Wenston

A local solvability result for operators with characteristics having odd order multiplicity

Abstract It is shown that an operator L with the canonical form L = Dt2p + 1 + a(t, Dx) is locally solvable if and only if a(t, Dx) satisfies a Nirenberg-Treves-type condition.

A necessary condition for the local solvability of the operator Pm2(x, D) + P2m − 1(x, D)

Abstract It is shown that a necessary condition for the local solvability of the operator P ( x , D ) = P m 2 ( x , D ) + P 2 m − 1 ( x , D ), where P m ( x , D ) is an m th-order homogeneous differential operator of principal type with real coefficients, is that along any null-bicharacteristic strip of P m ( x , ξ ) the imaginary part of the sub-principal symbol cannot have an odd-order zero where its real part does not vanish.

On Two Nonlinear Biharmonic Evolution Equations: Existence, Uniqueness and Stability

We study the following two nonlinear evolution equations with a fourth order (biharmonic) leading term: and with an initial value and a Dirichlet boundary conditions. We show the existence and uniqueness of the weak solutions of these two equations. For any t ∈ [0, + ∞ ), we prove that both solutions are in . We also discuss the asymptotic behavior of the solutions as time goes to infinity. This work lays the ground for our numerical simulations for the above systems [M.J. Lai, C. Liu and P. Wenston (2004). Numerical Simulations on Two Nonlinear Biharmonic Evolution Equations. Applicable Analysis, 83, 563–577].

L 1 Spline Methods for Scattered Data Interpolation and Approximation.

We generalize the L1 spline methods proposed in [4, 5] for scattered data interpolation and fitting using bivariate spline spaces of any degree d and any smoothness r (of course, r<d) over any triangulation. Some numerical experiments are presented to illustrate the better performance of the L1 spline methods as compared to the minimal energy method. We include some extensions for dealing with other surface design problems.

Bivariate spline method for numerical solution of steady state Navier–Stokes equations over polygons in stream function formulation

We use a bivariate spline method to solve the time evolution Navier-Stokes equations numerically. The bivariate splines we use in this article are in the spline space of smoothness r and degree 3r over triangulated quadrangulations. The stream function formulation for the Navier-Stokes equations is employed. Galerkins method is applied to discretize the space variables of the nonlinear fourth-order equation, Crank-Nicholsons method is applied to discretize the time variable, and Newtons iterative method is then used to solve the resulting nonlinear system. We show the existence and uniqueness of the weak solution in L2(0, T; H2(Ω)) ∩ L∞(0, T; H1(Ω)) of the 2D nonlinear fourth-order problem and give an estimate of how fast the numerical solution converges to the weak solution. The C1 cubic splines are implemented in MATLAB for solving the Navier-Stokes equations numerically. Our numerical experiments show that the method is effective and efficient.

Numerical Simulations on Two Nonlinear Biharmonic Evolution Equations

We numerically simulate the following two nonlinear evolution equations with a fourth-order (biharmonic) leading term: and with an initial value and a Dirichlet boundary conditions. We use a bivariate spline space like finite element method to solve these equations. We discuss the convergence of our numerical scheme and present several numerical experiments under different boundary conditions and different domains in the bivariate setting.

A sufficient condition for the local solvability of a linear partial differential operator with double characteristics

It is shown that a sufficient condition for the local solvability of an operator P(x, Dx) = Pm2 + P2m − 1, where P is an mth-order homogeneous operator of principal type with real coefficients, is that the imaginary part of the subprincipal symbol of P has constant sign near any null-bicharacteristic curve of Pm, and that the real part of the subprincipal symbol is not equal to zero when the imaginary part is.

On Schwarz's domain decomposition methods for elliptic boundary value problems

Summary. We study the additive and multiplicative Schwarz domain decomposition methods for elliptic boundary value problem of order 2 r based on an appropriate spline space of smoothness

Uniqueness in the Cauchy problem for weakly hyperbolic operators not satisfying Levi conditions

r-1

A well-posed boundary value problem for partial differential operators of principal type

. The finite element method reduces an elliptic boundary value problem to a linear system of equations. It is well known that as the number of triangles in the underlying triangulation is increased, which is indispensable for increasing the accuracy of the approximate solution, the size and condition number of the linear system increases. The Schwarz domain decomposition methods will enable us to break the linear system into several linear subsystems of smaller size. We shall show in this paper that the approximate solutions from the multiplicative Schwarz domain decomposition method converge to the exact solution of the linear system geometrically. We also show that the additive Schwarz domain decomposition method yields a preconditioner for the preconditioned conjugate gradient method. We tested these methods for the biharmonic equation with Dirichlet boundary condition over an arbitrary polygonal domain using