| Summary: | Let u be the solution to a general boundary value problem which is fourth order in the one-dimension space variable x. We consider various dependencies in the time variable t. We define a finite element approximation U to u. Let h, (DELTA)t and r be the space and time steps of the partition and the degree of the polynomials, respectively, used in the finite element method. We derive error estimates for U-u which are optimal in the power of h and, in the time independent case only, the Sobolev norm.
In the time independent case we derive the following optimal L('2) and L('(INFIN)) error estimates.
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In addition, we define super convergent approximations to u and its first three derivatives at an arbitrary point and show the error estimate to be O(h('2s-4)) for 2 (LESSTHEQ) s (LESSTHEQ) r + 1.
In the time dependent case we derive continuous time and discrete time error estimates for three problems. The first problem is second order in time with mixed boundary conditions. For 2 (LESSTHEQ) s (LESSTHEQ) r + 1, we show that
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The second problem is also second order in time, but in self-adjoint form with zero boundary conditions. For 2 (LESSTHEQ) s (LESSTHEQ) r + 1, we show that
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The third problem is first order in time with mixed boundary conditions. For 2 (LESSTHEQ) s (LESSTHEQ) 4 + 1, we show that
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All the above estimates which do not contain t-derivatives have optimal rate.
We define discrete time finite difference schemes for the above problems. We derive error estimates analogous to the above results with the addition of showing convergence is O(((DELTA)t)('2)).
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