Abstract
In this paper, we provide a new type of study approach for the two-dimensional (2D) Sobolev equations. We first establish a semi-discrete Crank-Nicolson (CN) formulation with second-order accuracy about time for the 2D Sobolev equations. Then we directly establish a fully discrete CN finite volume element (CNFVE) formulation from the semi-discrete CN formulation about time and provide the error estimates for the fully discrete CNFVE solutions. Finally, we provide a numerical example to verify the correction of theoretical conclusions. Further, it is shown that the fully discrete CNFVE formulation is better than the fully discrete FVE formulation with first-order accuracy in time.
Highlights
The finite volume element (FVE) method is a very effective discretization tool for the two-dimensional ( D) Sobolev equations
Whereas the fully discrete CN finite volume element (CNFVE) formulation here is directly established from the semi-discrete CN formulation about time, which avoids the semi-discrete FVE formulation with respect to spatial variables, such that the procedure of theoretical analysis for the fully discrete CNFVE formulation becomes simpler and more convenient than the classical FVE methods, which is a new type of study approach for the D Sobolev equations
4 A numerical example we provide a numerical example of the D Sobolev equations to illustrate that the numerical results are consistent with theoretical conclusions
Summary
The finite volume element (FVE) method (see [ , ]) is a very effective discretization tool for the two-dimensional ( D) Sobolev equations. In this paper, we establish a fully discrete CNFVE formulation with the second-order accuracy in time for the D Sobolev equations and provide the error estimates of the fully discrete CNFVE solutions. Whereas the fully discrete CNFVE formulation here is directly established from the semi-discrete CN formulation about time, which avoids the semi-discrete FVE formulation with respect to spatial variables, such that the procedure of theoretical analysis for the fully discrete CNFVE formulation becomes simpler and more convenient than the classical FVE methods, which is a new type of study approach for the D Sobolev equations. In Section , we establish the semi-discrete CN formulation about time for the D Sobolev equations and provide the error estimates of solutions for the formulation. √ ε}, which gets and completes the proof of Theorem
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