First, they reduce the globally coupled unknowns to the approximate trace of the solution on element boundaries, thereby leading to a significant reduction in the degrees of freedom.
Let BMOL(Rn) denote the BMO space on Rn associated to the Schrödinger operator L. In this article we will show that a function f∈BMOL(Rn) is the trace of the solution of Lu="−utt+Lu= 0, u x,0)="f(x), where u satisfies a Carleson conditionsupxB,rBr−nBrB∫0∫B xB,rB t|∇u x,t)|2dxdt⩽C<∞.
In the course of the analysis we establish the pointwise convergence of the trace of the solution along a path of extremum.
where u h is the solution of system (4.7 - 4.11) and u is the trace of the solution of (2.1), (2.2) on G ¯ ∗ h, n.
Let v be the trace of the solution of problem (13), (14) on (overline{D}_{h}^{prime}), and (v_{h}) be the solution of system (15), (16).
Let epsilon_{h}=u_{h}-u, (36) where (u_{h}) is the solution of system (8 - 11), and u is the trace of the solution of problem (1), (2) on (overline{D} _{ast}^{h,n}).
Proof Let ϵ h = u h − u, where u is the trace of the solution of problem (2.1), (2.2) on Π ¯ h, and u h is the solution of system (2.6 - 2.8 2.6 - 2.8
So, not only is Definition 1.1 in what follows different from the definitions of entropy solutions in [15, 17], but the trace of the solution in our paper is also in the traditional way.
If the diffusion coefficient of the equation is degenerate on the boundary, no matter we can define the trace of the solution on the boundary or not, by choosing a suitable test function, the stability of the solutions always can be established without a boundary condition.
