Abstract
We present full discretization of the telegraph equation with nonlocal coeffecient using Rothe-nite element method. For solving the equation numerically we use the Newton Raphson method, but the nonlocal term causes diffeculties because the Jacobien matrix is full. To remedy these diffeculties we apply the technique used by Sudhakar [4]. The optimal a priori error estimates for both semi discrete and fully discrete schemes are derived in V and H1 and a numerical experiment is described to support our theoretical result.
Highlights
Let u−1 be defined as u−1(x) = u0(x) − τ u1(x), the recurrent approximation scheme for i = 1, ..., n becomes
0 ≤ i ≤ n, we consider a triangulation Υih made of triangles T i such that no nodes of every triangle lies in the interior of a side of another triangle
Let Vhi be the discrete space of V i defined by
Summary
Let u−1 be defined as u−1(x) = u0(x) − τ u1(x), the recurrent approximation scheme for i = 1, ..., n becomes. ∼= u(., ti) ∈ V, i = 1, 2, ..., n, such that, δ2ui, v + δui, v + a(l(ui)) ui, v A = f i, v We define the Roth’s functions by a piecewise linear interpolation with respect to the time t,. − [0,T ] δun, ∂tv − δun(0), v(., 0) + [0,T ] ∂tun, v + [0,T ] a(l(un)) un, v A = [0,T ] fn, v (2.7). Choose v = δui in the equation (2.1), we get (δui − δui−1, δui) + τ δui 2 + m(ui, ui − ui−1)A ≤ τ f i δui.
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