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This paper studies three different equations of non- local boundary value and initial conditions in the first chapter of the introduction , a brief introduction to the application of non - local problem , current research and to study the main issues second chapter lists prior knowledge of the needs in this article . third chapter is the focus of this paper , using the finite element method , followed by solving the first two types of equations , first discuss the homogeneous elliptic equations ( Note : here is different from the generic sense homogeneous homogeneous ) To this end, the paper constructs H * space , and the corresponding finite element space . further introduction of the projection operator , and then , under certain assumptions , generalized the Lax - Milgram theorem was established , thus proving the well-posedness of the problem . In this paper, classical interpolation function error analysis theory, the optimal order of error of H * space interpolation function for non- homogeneous elliptic equations , by constructing a function that satisfy certain conditions , which will be non - homogeneous problem into homogeneous parabolic equations with source terms that p ( t ) = 0 , In this paper, a semi-discrete , fully discrete format , as well as the corresponding finite element error estimates . fourth chapter , the use of the principle of superposition for decomposition , discussion of parabolic equations with source terms , and gives specific discrete format and the corresponding error estimates .
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