Finite element methods for parabolic and hyperbolic partial integro- differential equations. (English) Zbl 0657.65142
For parabolic integro-partial differential equations of the following forms
\[
D_ tu-\nabla \cdot (a(x,t,u)\nabla u)=\int^{t}_{0}f(x,t,s,u(x,s),\nabla u(x,s))ds,\quad (x,t)\in \Omega \times (0,T],\quad and
\]
\[ C(x,t,u)D_ tu- D_{xx}u=\int^{t}_{0}f(x,t,s,u(x,s),D_ xu(x,s))ds,\quad (x,t)\in (0,1)\times (0,T], \] the authors formulate and investigate in detail several discrete-time Galerkin methods and discrete-time collocation methods, respectively, which are proved to provide approximations of second order in time and of optimal order of accuracy in space. Besides it they briefly describe Galerkin and collocation methods for hyperbolic integro-partial differential equations and give corresponding error estimates without proof.
\[ C(x,t,u)D_ tu- D_{xx}u=\int^{t}_{0}f(x,t,s,u(x,s),D_ xu(x,s))ds,\quad (x,t)\in (0,1)\times (0,T], \] the authors formulate and investigate in detail several discrete-time Galerkin methods and discrete-time collocation methods, respectively, which are proved to provide approximations of second order in time and of optimal order of accuracy in space. Besides it they briefly describe Galerkin and collocation methods for hyperbolic integro-partial differential equations and give corresponding error estimates without proof.
Reviewer: V.Kamen
Keywords:
parabolic integro-partial differential equations; discrete-time Galerkin methods; discrete-time collocation methods; optimal order of accuracy; hyperbolic integro-partial differential equations; error estimatesReferences:
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