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\(L^\infty(L^2)\) and \(L^\infty(L^\infty)\) error estimates for mixed methods for integro-differential equations of parabolic type. (English) Zbl 0941.65143
Studies in error analysis and quantitative estimates form an integral part of numerical investigations in applied mathematics. Many important phenomena in nature can be usually formulated with the help of suitable mathematical models. In order to get an insight into the effectiveness of various methods for treating different kinds of mathematical equations: differential, integral, integro-differential, etc., some new and efficient approaches are available now.
The author, in this 16-page paper, derives estimates for a mixed finite element method for the initial-boundary value problems stated in terms of the typical parabolic integro-differential equation: \[ u_t= \text{div}\Biggl\{{\mathfrak a}\cdot\nabla u+ \int^t_0{\mathfrak b}\cdot\nabla u d\tau+ \int^t_0 cu d\tau\Biggr\}+ f. \] Finding the introduction in the first section, questions of existence and uniqueness of solutions are discussed. Then a generalized mixed elliptic projection is set up in the third section, which has five theorems in it with proofs.
In the final section, optimal \(L^\infty\) \((L^2)\) error estimates and quasi-optimal error estimates for continuous in-time mixed finite element approximation are obtained.

65R20 Numerical methods for integral equations
45K05 Integro-partial differential equations
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