Sentence examples for both the scalar variable from inspiring English sources

Mixed finite element methods are appropriate for the state equations in such cases since both the scalar variable and its flux variable can be approximated to the same accuracy by using such methods; see, for example, [20 23].

When the objective functional contains the gradient of the state variable, mixed finite element methods should be used for discretization of the state equation with which both the scalar variable and its flux variable can be approximated in the same accuracy.

First, we express the approximate scalar variable and corresponding flux within each element in terms of an approximate trace of the scalar variable along the element boundary.

The methods are devised by expressing the approximate scalar variable and corresponding flux in terms of an approximate trace of the scalar variable and then explicitly enforcing the jump condition of the numerical fluxes across the element boundary.

Finally, we apply element-by-element postprocessing schemes to obtain new approximations of the flux and the scalar variable.

Indeed, when the time-marching method is (p+1 th order accurate and when polynomials of degree p⩾0 are used to represent the scalar variable, the flux and the approximate trace, we observe that the approximations for the scalar variable, the flux and the trace of the scalar variable converge with the optimal order of p+1 in the L2-norm.

For higher orders schemes, we observed super-convergence by one order for the scalar variable which is consistent with the previously published result for a symmetric diffusion tensor.

Finally, we introduce a simple element-by-element postprocessing scheme to obtain new approximations of the flux and the scalar variable.

When the time-marching method is (p+1 th order accurate and when polynomials of degree p⩾0 are used to represent the scalar variable, each component of the flux and the approximate trace, we observe that the approximations for the scalar variable and the flux converge with the optimal order of p+1 in the L2-norm.

We present and analyze a first order least squares method for convection dominated diffusion problems, which provides robust L2 a priori error estimate for the scalar variable even if the given data f∈L2.

The scalar variable D, which characterises the material damage, is written as a function of the life duration β, using a modified form of the Mankowsky empirical law [Int J Solids Struct 32(1995 1607] 1995 1607]