| Summary: | We study the backward Euler fully discrete mixed finite element method for the time-dependent Schrödinger equation; the error result of the mixed finite element solution is obtained in the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>2</mn></msup></semantics></math></inline-formula>-norm with order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo>(</mo><mi>τ</mi><mo>+</mo><msup><mi>h</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></semantics></math></inline-formula>. Then, a two-grid method is presented with a backward Euler fully discrete scheme. Using this method, we solve the original problem on a much coarser grid and solve elliptic equations on a fine grid. In addition, the error of the two-grid solution is also obtained in the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>2</mn></msup></semantics></math></inline-formula>-norm with order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo>(</mo><mi>τ</mi><mo>+</mo><msup><mi>h</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>+</mo><msup><mi>H</mi><mrow><mi>k</mi><mo>+</mo><mn>2</mn></mrow></msup><mo>)</mo></mrow></semantics></math></inline-formula>. The numerical experiment is provided to demonstrate the efficiency of the algorithm.
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