Control of the local magnetic fields desirable for spintronics and quantum information technology is not well developed. Existing methods produce either moderately small local fields or one field orientation. We present designs of patterned magnetic elements that produce remanent fields of 50 mT (potentially 200 mT) confined to chosen, submicrometer regions in directions perpendicular to an external initializing field. A wide variety of magnetic-field profiles on nanometer scales can be produced with the option of applying electric fields, for example, to move a quantum dot between regions where the magnetic-field direction or strength is different. We have confirmed our modeling by measuring the fields in one design using electron holography.