Forcing schema objects in a new file group?

Yes, there is a way to do exactly what you want with a DDL trigger. This one ensures objects in the dbo schema store their data in TestPartitionGroup1.

CREATE TRIGGER SchemaBasedFilegroupLock 
ON DATABASE 
FOR CREATE_TABLE, CREATE_INDEX, ALTER_INDEX
AS 
    DECLARE @schemaName [sysname]
    DECLARE @allowedDataSpaceName [sysname]

    SET @schemaName = 'dbo'
    SET @allowedDataSpaceName = 'TestPartitionGroup1'

    DECLARE @objectName [sysname]

    SELECT @objectName = 
        CASE WHEN EVENTDATA().value('(/EVENT_INSTANCE/ObjectType)[1]','nvarchar(max)') = 'INDEX'
        THEN
            QUOTENAME(EVENTDATA().value('(/EVENT_INSTANCE/SchemaName)[1]','nvarchar(max)')) + '.' + 
            QUOTENAME(EVENTDATA().value('(/EVENT_INSTANCE/TargetObjectName)[1]','nvarchar(max)'))                
        ELSE
            QUOTENAME(EVENTDATA().value('(/EVENT_INSTANCE/SchemaName)[1]','nvarchar(max)')) + '.' + 
            QUOTENAME(EVENTDATA().value('(/EVENT_INSTANCE/ObjectName)[1]','nvarchar(max)'))
        END

    IF OBJECT_SCHEMA_NAME(OBJECT_ID(@objectName)) = @schemaName AND EXISTS 
    (SELECT * FROM [sys].[indexes] WHERE [object_id] = OBJECT_ID(@objectName) AND [data_space_id] != 
    (SELECT [data_space_id] FROM [sys].[data_spaces] WHERE [sys].[data_spaces].[name] = @allowedDataSpaceName))
    BEGIN
      DECLARE @errorMessage [nvarchar](MAX)
      SET @errorMessage = 'Storage objects in the ''' + @schemaName + ''' schema can only place their data in the ''' + @allowedDataSpaceName + ''' data space.'
      RAISERROR (@errorMessage, 16, 1)      
      ROLLBACK
    END;

Your best bet is through training and policy - if you are looking for a default to force just the new schema objects to be forced to the new filegroup.

Setting default filegroup will affect all new objects and force them to be created on the default filegroup.

You would therefore need to specify the filegroup at creation time of the new object.

In addition to setting the defualt filegroup as CirqueDeSQLeil says, you can alter the user to give them a default schema. This example is from Books Online:

USE AdventureWorks;
ALTER USER Mary51 WITH DEFAULT_SCHEMA = Purchasing;
GO

But, just because you have a default schema doesn't mean that you can't override that schema when you create an object if that user has access to the ability to create objects on more than one schema.