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Introduction"},{"canvases":["https://noah.nrw/i3f/v20/6249177/canvas/6313891","https://noah.nrw/i3f/v20/6249177/canvas/6313892","https://noah.nrw/i3f/v20/6249177/canvas/6313893"],"@id":"https://noah.nrw/i3f/v20/6249177/range/6313890","@type":"sc:Range","label":"2. Notation"},{"canvases":["https://noah.nrw/i3f/v20/6249177/canvas/6313895","https://noah.nrw/i3f/v20/6249177/canvas/6313897","https://noah.nrw/i3f/v20/6249177/canvas/6313899","https://noah.nrw/i3f/v20/6249177/canvas/6313900"],"ranges":["https://noah.nrw/i3f/v20/6249177/range/6313896","https://noah.nrw/i3f/v20/6249177/range/6313898"],"@id":"https://noah.nrw/i3f/v20/6249177/range/6313894","@type":"sc:Range","label":"3. Basic idea applied to 1-D obstacle problem"},{"canvases":["https://noah.nrw/i3f/v20/6249177/canvas/6313897"],"@id":"https://noah.nrw/i3f/v20/6249177/range/6313896","@type":"sc:Range","label":"3.1. Possible variational formulations"},{"canvases":["https://noah.nrw/i3f/v20/6249177/canvas/6313899","https://noah.nrw/i3f/v20/6249177/canvas/6313900"],"@id":"https://noah.nrw/i3f/v20/6249177/range/6313898","@type":"sc:Range","label":"3.2. Transformation into the system of equations"},{"canvases":["https://noah.nrw/i3f/v20/6249177/canvas/6313903","https://noah.nrw/i3f/v20/6249177/canvas/6313904","https://noah.nrw/i3f/v20/6249177/canvas/6313906","https://noah.nrw/i3f/v20/6249177/canvas/6313907","https://noah.nrw/i3f/v20/6249177/canvas/6313909","https://noah.nrw/i3f/v20/6249177/canvas/6313910","https://noah.nrw/i3f/v20/6249177/canvas/6313911","https://noah.nrw/i3f/v20/6249177/canvas/6313913","https://noah.nrw/i3f/v20/6249177/canvas/6313914","https://noah.nrw/i3f/v20/6249177/canvas/6313915","https://noah.nrw/i3f/v20/6249177/canvas/6313916","https://noah.nrw/i3f/v20/6249177/canvas/6313917","https://noah.nrw/i3f/v20/6249177/canvas/6313919","https://noah.nrw/i3f/v20/6249177/canvas/6313920","https://noah.nrw/i3f/v20/6249177/canvas/6313921","https://noah.nrw/i3f/v20/6249177/canvas/6313922","https://noah.nrw/i3f/v20/6249177/canvas/6313923","https://noah.nrw/i3f/v20/6249177/canvas/6313924","https://noah.nrw/i3f/v20/6249177/canvas/6313925","https://noah.nrw/i3f/v20/6249177/canvas/6313926","https://noah.nrw/i3f/v20/6249177/canvas/6313927","https://noah.nrw/i3f/v20/6249177/canvas/6313928","https://noah.nrw/i3f/v20/6249177/canvas/6313929","https://noah.nrw/i3f/v20/6249177/canvas/6313930","https://noah.nrw/i3f/v20/6249177/canvas/6313932","https://noah.nrw/i3f/v20/6249177/canvas/6313933","https://noah.nrw/i3f/v20/6249177/canvas/6313934","https://noah.nrw/i3f/v20/6249177/canvas/6313935","https://noah.nrw/i3f/v20/6249177/canvas/6313936","https://noah.nrw/i3f/v20/6249177/canvas/6313937","https://noah.nrw/i3f/v20/6249177/canvas/6313939","https://noah.nrw/i3f/v20/6249177/canvas/6313940","https://noah.nrw/i3f/v20/6249177/canvas/6313941","https://noah.nrw/i3f/v20/6249177/canvas/6313942","https://noah.nrw/i3f/v20/6249177/canvas/6313943","https://noah.nrw/i3f/v20/6249177/canvas/6313944","https://noah.nrw/i3f/v20/6249177/canvas/6313945","https://noah.nrw/i3f/v20/6249177/canvas/6313946","https://noah.nrw/i3f/v20/6249177/canvas/6313947","https://noah.nrw/i3f/v20/6249177/canvas/6313948","https://noah.nrw/i3f/v20/6249177/canvas/6313949","https://noah.nrw/i3f/v20/6249177/canvas/6313950"],"ranges":["https://noah.nrw/i3f/v20/6249177/range/6313902","https://noah.nrw/i3f/v20/6249177/range/6313905","https://noah.nrw/i3f/v20/6249177/range/6313908","https://noah.nrw/i3f/v20/6249177/range/6313912","https://noah.nrw/i3f/v20/6249177/range/6313918","https://noah.nrw/i3f/v20/6249177/range/6313931","https://noah.nrw/i3f/v20/6249177/range/6313938"],"@id":"https://noah.nrw/i3f/v20/6249177/range/6313901","@type":"sc:Range","label":"4. Stokes problem with cavitation eects"},{"canvases":["https://noah.nrw/i3f/v20/6249177/canvas/6313903","https://noah.nrw/i3f/v20/6249177/canvas/6313904"],"@id":"https://noah.nrw/i3f/v20/6249177/range/6313902","@type":"sc:Range","label":"4.1. Minimization problem and equivalent variational formulation"},{"canvases":["https://noah.nrw/i3f/v20/6249177/canvas/6313906","https://noah.nrw/i3f/v20/6249177/canvas/6313907"],"@id":"https://noah.nrw/i3f/v20/6249177/range/6313905","@type":"sc:Range","label":"4.2. First proposal for solution strategy"},{"canvases":["https://noah.nrw/i3f/v20/6249177/canvas/6313909","https://noah.nrw/i3f/v20/6249177/canvas/6313910","https://noah.nrw/i3f/v20/6249177/canvas/6313911"],"@id":"https://noah.nrw/i3f/v20/6249177/range/6313908","@type":"sc:Range","label":"4.3. First numerical results"},{"canvases":["https://noah.nrw/i3f/v20/6249177/canvas/6313913","https://noah.nrw/i3f/v20/6249177/canvas/6313914","https://noah.nrw/i3f/v20/6249177/canvas/6313915","https://noah.nrw/i3f/v20/6249177/canvas/6313916","https://noah.nrw/i3f/v20/6249177/canvas/6313917"],"@id":"https://noah.nrw/i3f/v20/6249177/range/6313912","@type":"sc:Range","label":"4.4. Further analysis of the possible solution strategies"},{"canvases":["https://noah.nrw/i3f/v20/6249177/canvas/6313919","https://noah.nrw/i3f/v20/6249177/canvas/6313920","https://noah.nrw/i3f/v20/6249177/canvas/6313921","https://noah.nrw/i3f/v20/6249177/canvas/6313922","https://noah.nrw/i3f/v20/6249177/canvas/6313923","https://noah.nrw/i3f/v20/6249177/canvas/6313924","https://noah.nrw/i3f/v20/6249177/canvas/6313925","https://noah.nrw/i3f/v20/6249177/canvas/6313926","https://noah.nrw/i3f/v20/6249177/canvas/6313927","https://noah.nrw/i3f/v20/6249177/canvas/6313928","https://noah.nrw/i3f/v20/6249177/canvas/6313929","https://noah.nrw/i3f/v20/6249177/canvas/6313930"],"@id":"https://noah.nrw/i3f/v20/6249177/range/6313918","@type":"sc:Range","label":"4.5. Numerical results"},{"canvases":["https://noah.nrw/i3f/v20/6249177/canvas/6313932","https://noah.nrw/i3f/v20/6249177/canvas/6313933","https://noah.nrw/i3f/v20/6249177/canvas/6313934","https://noah.nrw/i3f/v20/6249177/canvas/6313935","https://noah.nrw/i3f/v20/6249177/canvas/6313936","https://noah.nrw/i3f/v20/6249177/canvas/6313937"],"@id":"https://noah.nrw/i3f/v20/6249177/range/6313931","@type":"sc:Range","label":"4.6. Existence and uniqueness of the continuous and the FEM solutions"},{"canvases":["https://noah.nrw/i3f/v20/6249177/canvas/6313939","https://noah.nrw/i3f/v20/6249177/canvas/6313940","https://noah.nrw/i3f/v20/6249177/canvas/6313941","https://noah.nrw/i3f/v20/6249177/canvas/6313942","https://noah.nrw/i3f/v20/6249177/canvas/6313943","https://noah.nrw/i3f/v20/6249177/canvas/6313944","https://noah.nrw/i3f/v20/6249177/canvas/6313945","https://noah.nrw/i3f/v20/6249177/canvas/6313946","https://noah.nrw/i3f/v20/6249177/canvas/6313947","https://noah.nrw/i3f/v20/6249177/canvas/6313948","https://noah.nrw/i3f/v20/6249177/canvas/6313949","https://noah.nrw/i3f/v20/6249177/canvas/6313950"],"@id":"https://noah.nrw/i3f/v20/6249177/range/6313938","@type":"sc:Range","label":"4.7. Error estimate"},{"canvases":["https://noah.nrw/i3f/v20/6249177/canvas/6313953","https://noah.nrw/i3f/v20/6249177/canvas/6313955","https://noah.nrw/i3f/v20/6249177/canvas/6313956","https://noah.nrw/i3f/v20/6249177/canvas/6313958","https://noah.nrw/i3f/v20/6249177/canvas/6313960","https://noah.nrw/i3f/v20/6249177/canvas/6313961","https://noah.nrw/i3f/v20/6249177/canvas/6313962","https://noah.nrw/i3f/v20/6249177/canvas/6313963"],"ranges":["https://noah.nrw/i3f/v20/6249177/range/6313952","https://noah.nrw/i3f/v20/6249177/range/6313954","https://noah.nrw/i3f/v20/6249177/range/6313957","https://noah.nrw/i3f/v20/6249177/range/6313959"],"@id":"https://noah.nrw/i3f/v20/6249177/range/6313951","@type":"sc:Range","label":"5. Revisiting Obstacle problem in 2-D"},{"canvases":["https://noah.nrw/i3f/v20/6249177/canvas/6313953"],"@id":"https://noah.nrw/i3f/v20/6249177/range/6313952","@type":"sc:Range","label":"5.1. Introduction of the mixed formulation"},{"canvases":["https://noah.nrw/i3f/v20/6249177/canvas/6313955","https://noah.nrw/i3f/v20/6249177/canvas/6313956"],"@id":"https://noah.nrw/i3f/v20/6249177/range/6313954","@type":"sc:Range","label":"5.2. Numerical treatment of the mixed obstacle problem"},{"canvases":["https://noah.nrw/i3f/v20/6249177/canvas/6313958"],"@id":"https://noah.nrw/i3f/v20/6249177/range/6313957","@type":"sc:Range","label":"5.3. Numerical results"},{"canvases":["https://noah.nrw/i3f/v20/6249177/canvas/6313960","https://noah.nrw/i3f/v20/6249177/canvas/6313961","https://noah.nrw/i3f/v20/6249177/canvas/6313962","https://noah.nrw/i3f/v20/6249177/canvas/6313963"],"@id":"https://noah.nrw/i3f/v20/6249177/range/6313959","@type":"sc:Range","label":"5.4. Error estimate"},{"canvases":["https://noah.nrw/i3f/v20/6249177/canvas/6313965","https://noah.nrw/i3f/v20/6249177/canvas/6313967","https://noah.nrw/i3f/v20/6249177/canvas/6313968","https://noah.nrw/i3f/v20/6249177/canvas/6313969","https://noah.nrw/i3f/v20/6249177/canvas/6313970","https://noah.nrw/i3f/v20/6249177/canvas/6313971","https://noah.nrw/i3f/v20/6249177/canvas/6313972","https://noah.nrw/i3f/v20/6249177/canvas/6313973","https://noah.nrw/i3f/v20/6249177/canvas/6313974","https://noah.nrw/i3f/v20/6249177/canvas/6313976","https://noah.nrw/i3f/v20/6249177/canvas/6313977","https://noah.nrw/i3f/v20/6249177/canvas/6313978","https://noah.nrw/i3f/v20/6249177/canvas/6313979","https://noah.nrw/i3f/v20/6249177/canvas/6313980","https://noah.nrw/i3f/v20/6249177/canvas/6313981","https://noah.nrw/i3f/v20/6249177/canvas/6313982","https://noah.nrw/i3f/v20/6249177/canvas/6313983","https://noah.nrw/i3f/v20/6249177/canvas/6313984","https://noah.nrw/i3f/v20/6249177/canvas/6313985","https://noah.nrw/i3f/v20/6249177/canvas/6313986"],"ranges":["https://noah.nrw/i3f/v20/6249177/range/6313966","https://noah.nrw/i3f/v20/6249177/range/6313975"],"@id":"https://noah.nrw/i3f/v20/6249177/range/6313964","@type":"sc:Range","label":"6. Framework for first kind problems with linear inequality conditions"},{"canvases":["https://noah.nrw/i3f/v20/6249177/canvas/6313967","https://noah.nrw/i3f/v20/6249177/canvas/6313968","https://noah.nrw/i3f/v20/6249177/canvas/6313969","https://noah.nrw/i3f/v20/6249177/canvas/6313970","https://noah.nrw/i3f/v20/6249177/canvas/6313971","https://noah.nrw/i3f/v20/6249177/canvas/6313972","https://noah.nrw/i3f/v20/6249177/canvas/6313973","https://noah.nrw/i3f/v20/6249177/canvas/6313974"],"@id":"https://noah.nrw/i3f/v20/6249177/range/6313966","@type":"sc:Range","label":"6.1. Optimization theory for functionals"},{"canvases":["https://noah.nrw/i3f/v20/6249177/canvas/6313976","https://noah.nrw/i3f/v20/6249177/canvas/6313977","https://noah.nrw/i3f/v20/6249177/canvas/6313978","https://noah.nrw/i3f/v20/6249177/canvas/6313979","https://noah.nrw/i3f/v20/6249177/canvas/6313980","https://noah.nrw/i3f/v20/6249177/canvas/6313981","https://noah.nrw/i3f/v20/6249177/canvas/6313982","https://noah.nrw/i3f/v20/6249177/canvas/6313983","https://noah.nrw/i3f/v20/6249177/canvas/6313984","https://noah.nrw/i3f/v20/6249177/canvas/6313985","https://noah.nrw/i3f/v20/6249177/canvas/6313986"],"@id":"https://noah.nrw/i3f/v20/6249177/range/6313975","@type":"sc:Range","label":"6.2. Newton-type method for the generalized problem"},{"canvases":["https://noah.nrw/i3f/v20/6249177/canvas/6313988","https://noah.nrw/i3f/v20/6249177/canvas/6313989"],"@id":"https://noah.nrw/i3f/v20/6249177/range/6313987","@type":"sc:Range","label":"7. Summary and outlook"},{"canvases":["https://noah.nrw/i3f/v20/6249177/canvas/6313991","https://noah.nrw/i3f/v20/6249177/canvas/6313993","https://noah.nrw/i3f/v20/6249177/canvas/6313994","https://noah.nrw/i3f/v20/6249177/canvas/6313995","https://noah.nrw/i3f/v20/6249177/canvas/6313996"],"ranges":["https://noah.nrw/i3f/v20/6249177/range/6313992"],"@id":"https://noah.nrw/i3f/v20/6249177/range/6313990","@type":"sc:Range","label":"Appendices"},{"canvases":["https://noah.nrw/i3f/v20/6249177/canvas/6313993","https://noah.nrw/i3f/v20/6249177/canvas/6313994","https://noah.nrw/i3f/v20/6249177/canvas/6313995","https://noah.nrw/i3f/v20/6249177/canvas/6313996"],"@id":"https://noah.nrw/i3f/v20/6249177/range/6313992","@type":"sc:Range","label":"A. Useful theorems and 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